Advances on strong colorings over partitions
aa r X i v : . [ m a t h . L O ] F e b ADVANCES ON STRONG COLORINGS OVER PARTITIONS
MENACHEM KOJMAN, ASSAF RINOT, AND JURIS STEPR ¯ANS
Abstract.
We advance the theory of strong colorings over partitions andobtain positive and negative results at the level of ℵ and at higher cardinals.Improving a strong coloring theorem due to Galvin [Gal80], we prove thatthe existence of a non-meager set of reals of cardinality ℵ is equivalent to thehigher dimensional version of an unbalanced negative partition relation dueto Erd˝os, Hajnal and Milner [EHM66]. We then prove that these coloringscannot be strengthened to overcome countable partitions of [ ℵ ] , even in thepresence of both a Luzin set and a coherent Souslin tree.A correspondence between combinatorial properties of partitions and chainconditions of natural forcing notions for destroying strong colorings over themis uncovered and allows us to prove positive partition relations for ℵ fromweak forms of Martin’s Axiom, thereby answering two questions from [CKS20].Positive partition relations for ℵ and higher cardinals are similarly deducedfrom the Generalized Martin’s Axiom.Finally, we provide a group of pump-up theorems for strong colorings overpartitions. Some of them solve more problems from [CKS20]. Introduction
Shortly after Ramsey [Ram30] proved his groundbreaking result that every infi-nite graph contains an infinite clique or an infinite anti-clique, Sierpi´nski [Sie33] de-fined a graph over the reals with neither an uncountable cliques nor an uncountableanti-cliques. Sierpi´nski’s negative partition relation suggested that there was a the-ory of strong colorings waiting to be discovered on the uncountable cardinals. Sur-veys of the rich theory of strong colorings that was developed since Sierpi´nski’s timeto the present time may be found in the introductions to [Rin14a, Rin14b, CKS20].We settle here, then, to mentioning only a few milestones.Sierpi´nski’s theorem gives rise to a coloring c : [ ℵ ] → c [[ A ] ] = 2 for all uncountable A ⊆ ℵ . The existence of such a coloring is assertedby the symbol ℵ [ ℵ ] .Improving Sierpi´nski’s theorem to handle a larger number of colors was very chal-lenging. After many years and considerable effort by many, Todorˇcevi´c constructedin [Tod87] a coloring c : [ ℵ ] → ℵ with the property that c [[ A ] ] = ℵ for alluncountable A ⊆ ℵ . The existence of such colorings is expressed by ℵ [ ℵ ] ℵ .A second way for making a strong coloring stronger is to require that it attainsall possible colors on additional graphs beyond squares , i.e., sets of the form [ A ] = { ( α, β ) ∈ A × A | α < β } . For instance, in [Moo06], Moore constructed a coloring c : [ ℵ ] → ℵ with the property that c [ A ⊛ B ] = ℵ for all uncountable A, B ⊆ ℵ ,where A ⊛ B stands for the rectangle { ( α, β ) ∈ A × B | α < β } . We denote this by ℵ [ ℵ ⊛ ℵ ] ℵ . Mathematics Subject Classification.
Primary 03E02; Secondary 03E35, 03E17.
Assuming the continuum hypothesis ( CH ), Erd˝os, Hajnal and Rado [EHR65,Theorem 17A] constructed a coloring c : [ ℵ ] → ℵ with the property that c [ A ⊛ B ] = ℵ for all infinite A ⊆ ℵ and uncountable B ⊆ ℵ , and then Erd˝os, Hajnaland Milner [EHM66, Lemma 14.1] used CH to construct a coloring c : [ ℵ ] → ℵ with the property that for all infinite A ⊆ ℵ and uncountable B ⊆ ℵ , thereis α ∈ A such that c [ { α } ⊛ B ] = ℵ . We denote the existence of the former by ℵ [ ℵ ⊛ ℵ ] ℵ , and of the latter by ℵ [ ℵ ⊛ ℵ (cid:30) ⊛ ℵ ] ℵ . Here, the class ofgraphs is enlarged and all colors are attained on a subgraph of some prescribedform.A third aspect in which a coloring may become stronger is its ability to handlesets of higher dimension. In [Gal80], Galvin constructed from CH a coloring c :[ ℵ ] → k , every uncountable pairwisedisjoint subfamily A ⊆ [ ℵ ] k , and every color γ <
2, there are a, b ∈ A withmax( a ) < min( b ) such that c [ a × b ] = { γ } . The instance k = 2 implies, as is seeneasily, that c witnesses ℵ [ ℵ ⊛ ℵ ] ℵ . In Shelah’s notation [She88], Galvin’stheorem is denoted by Pr ( ℵ , ℵ , , ℵ ). A careful look at Galvin’s proof revealsthat CH gives, if fact, a higher-dimensional version of the Erd˝os-Hajnal-Milnertheorem, which we denote by Pr ( ℵ , ℵ ⊛ ℵ (cid:30) ⊛ ℵ , ℵ , ℵ ).Motivated by Roitman’s work [Roi78] which connected strong colorings andtopology, Shelah [She88] identified a further aspect of strengthening a coloring,in which instead of requiring c ↾ ( a × b ) to be a constant function with someprescribed value γ , one requires c ↾ ( a × b ) to realize some arbitrary prescribedfinite pattern g . For instance, Pr ( ℵ , ℵ , θ, ℵ ) asserts the existence of a coloring c : [ ℵ ] → θ such that for every finite dimension k , every uncountable pairwisedisjoint subfamily A ⊆ [ ℵ ] k , and every pattern g : k × k → θ , there are a, b ∈ A with max( a ) < min( b ) such that c ( a ( i ) , b ( j )) = g ( i, j ) for all i, j < k .Finally — and this is the subject matter of the present article — we may aska coloring to overcome a certain partition of [ ℵ ] into countably many pieces.That is, given a partition p : [ ℵ ] → ℵ , we say that a Sierpi´nski-type coloring c : [ ℵ ] → θ is strong over p if for every uncountable A ⊆ ℵ there is some cell j < ω (depending on A ) such that c [[ A ] ∩ p − [ { j } ]] = θ . That is, c attains allcolors on pairs from A , even when restricted to some single p -cell. We denote thisby ℵ p [ ℵ ] θ .“Strong over p ” versions of each of the concepts above are found in Definition 2.2below. In each of these definitions, every coloring which is strong over p is alsostrong over any partition p ′ coarser than p . In particular, a strong coloring withouta partition is the same as a strong coloring over the trivial partition with one cell.To illustrate all different strenthenings we reviewed above in a single instance letus parse now a (consistent) strong coloring assertion in which all appear. For apartition p : [ ℵ ] → ℵ , Pr ( ℵ , ℵ ⊛ ℵ (cid:30) ⊛ ℵ , ℵ , ℵ ) p asserts the existence of a coloring c : [ ℵ ] → ℵ such that for all k, l < ω andpairwise disjoint subfamilies A ⊆ [ ℵ ] k , B ⊆ [ ℵ ] l with |A| = ℵ and |B| = ℵ , thereis a ∈ A such that for every matrix { τ i,j | i < k, j < l } of functions from ℵ to ℵ ,there is b ∈ B with max( a ) < min( b ) for which ^ i The theory of strong colorings over partitions generalizes the classical theory ofstrong colorings and offers several new points of view on the subject which, as weshall see below, contribute to the development of the classical theory.In [CKS20], it was proved that whether ℵ p [ ℵ ] ℵ holds for all partitions p : [ ℵ ] → ℵ is independent of ZFC . Specifically, if either CH holds or ℵ manyCohen reals have been added to any model of ZFC , ℵ p [ ℵ ] ℵ holds for all such p ; on the other hand, there is a forcing extension of the universe in which thisstatement fails with some generic p . Thus, the validity of the “over p ” version ofknown strong coloring relations is a meaningful question which is sensitive to theaxioms of set theory.In the present article, the theory of both negative and positive Ramsey-theoreticassertions over partitions is broadened considerably beyond the initial results of[CKS20]. Problems which were raised there about forcing axioms are now satisfac-torily solved, as a result of a good understanding of the correspondence betweencombinatorial properties of a partition and the forcing notions naturally associatedto it for handling strong colorings over it. Pump-up problems from [CKS20] arealso solved in satisfactory genrality with a new method for streching colors andstrengthening colorings over partitions. Finally, also the classical theory of strongcolorings over ℵ benefits from the techniques which are developed below, with animprovement of Galvin’s coloring theorem from [Gal80].1.1. Roadmap. Section 2 introduces the fundamental concepts, and establishessome preliminary results.Section 3 belongs almost entirely to the classical theory. In [Tod87], Todorˇcevi´cshows that Sierpi´nski’s Onto Mapping principle (to be defined in that section)and the Erd˝os-Hajnal-Milner unbalanced relation ℵ [ ℵ ⊛ ℵ (cid:30) ⊛ ℵ ] ℵ imply eachother and follow from the existence of a Luzin set. Recently, Miller [Mil14] andGuzm´an [Guz17] showed that Sieprinski’s principle is equivalent to non( M ) = ℵ ,the existence of a non-meager subset of the real line of cardinality ℵ . By a theoremof Shelah [She80], the existence of a Luzin set is strictly stronger than non( M ) = ℵ .In this section, we expand the circle of statements which are equivalent tonon( M ) = ℵ and add to it a higher dimensional version of the Sierpi´nski OntoMapping principle and the higher dimensional unbalanced strong colorings ´a laErd˝os-Hajnal-Milner, even over some partitions. We isolate now two out of the cir-cle of 7 equivalent statements considered in Section 3. As mentioned earlier, theirequivalence improves Galvin’s result from [Gal80]: Theorem A. non ( M ) = ℵ iff Pr ( ℵ , ℵ ⊛ ℵ (cid:30) ⊛ ℵ , ℵ , ℵ ) p holds for every ℓ ∞ -coherent partition p : [ ℵ ] → ℵ . (cid:3) We shall see later that Theorem A cannot be extended to cover all partitions.For this, we first turn to study the possible failures of Pr i ( ℵ , . . . ) p , such as thefailure of the simplest instance, ℵ p [ ℵ ] .For a partition p : [ κ ] → µ , denote by κ → p [ κ ] θ,<θ ′ the assertion that for everycoloring c : [ κ ] → θ , there is X ⊆ κ of size κ such that, for any cell j < µ , |{ c ( α, β ) | ( α, β ) ∈ [ X ] & p ( α, β ) = j }| < θ ′ . The smaller the θ ′ , the stronger the failure of κ p [ κ ] θ . The extreme case κ → p [ κ ] θ,< is denoted by κ → p ( κ ) θ . It asserts that for every coloring c : [ κ ] → θ ,there exists X ⊆ κ of size κ which is ( p, c ) -homogeneous , that is, the color of M. KOJMAN, A. RINOT, AND J. STEPR¯ANS ( α, β ) ∈ [ X ] is determined by the p -cell of ( α, β ): there is a function τ : µ → θ such that c ( α, β ) = τ ( p ( α, β )) for all ( α, β ) ∈ [ X ] .Following [LS77], for a partition p : [ κ ] → µ and a coloring c : [ κ ] → θ , we let z p ( c ) := min { ζ ≤ κ | κ may be covered by ζ many ( p, c )-homogeneous sets } . A very strong failure of κ p [ κ ] θ is the assertion that z p ( c ) < κ for every coloring c : [ κ ] → θ .In [CKS20], it was shown consistent to have a partition p : [ ℵ ] → ℵ forwhich ℵ → p ( ℵ ) holds, and consistent to have a partition p : [ ℵ ] → ℵ forwhich ℵ → p [ ℵ ] ℵ ,< ℵ holds. In both models from [CKS20], the partitions wereadded generically and then an iteration was carried out to “kill” every potentialstrong coloring over p by introducing an uncountable set on which all colors are notattained in any single p -cell.Section 4 improves the above-mentioned consistency results simultaneously. Itestablishes the consistency of z p ( c ) ≤ ℵ for all c : [ ℵ ] → ℵ for concrete partitions p . In particular, it is consistent that ℵ → p ( ℵ ) ℵ holds for some partition p . Aclose correspondence between the combinatorial properties of a partition p : [ κ ] → µ and chain conditions of the natural poset for forcing z p ( c ) < κ is uncovered,which enables our new proof strategy in which generic partitions are replaced bycombinatorially suitable ones. A similar correspondence enables us to establish theconsistency of κ → p ( κ ) θ for cardinals κ > ℵ : Theorem B. (1) If GMA λ + holds for a cardinal λ <λ = λ then there is a par-tition p : [ λ + ] → λ such that for every coloring c : [ λ + ] → λ , there is adecomposition λ + = U i<λ X i such that for all i, j < λ , |{ c ( α, β ) | ( α, β ) ∈ [ X i ] & p ( α, β ) = j }| = 1 . (2) If MA ℵ ( K ) holds then for every partition p : [ ℵ ] → ℵ with finite-to-onefibers, for every coloring c : [ ℵ ] → ℵ there is decomposition ℵ = U i<ω X i such that for all i, j < ω , { c ( α, β ) | ( α, β ) ∈ [ X i ] & p ( α, β ) = j } is finite . (3) Assuming MA ℵ ( K ) , for every partition p : [ ℵ ] → ℵ , the following areequivalent: (a) ℵ → p [ ℵ ] ℵ ,< ℵ ; (b) There exists X ∈ [ ℵ ] ℵ such that p ↾ [ X ] witnesses U( ℵ , ℵ , ℵ , ℵ ) .Remark. GMA λ + stands for the Generalized Martin’s Axiom at the level of λ + , and MA ℵ ( K ) stands for Martin’s Axiom for partial orders having the Knaster property.Their definitions may be found in Section 4. Note that MA ℵ ( K ) implies GMA ℵ ,so, in particular, MA ℵ ( K ) entails ℵ → p ( ℵ ) ℵ for some partition p : [ ℵ ] → ℵ .U( ℵ , ℵ , ℵ , ℵ ) is a provable instance of the 4-parameter principle U( . . . ) due toLambie-Hanson and Rinot [LR18]. Its definition is reproduced in Section 2.Section 5 fulfills our promise that Theorem A cannot be extended to all countablepartitions: Theorem C. It is consistent with the existence of both a Luzin set and a coherentSouslin tree that there is a partition p : [ ℵ ] → ℵ such that z p ( c ) ≤ ℵ for everycoloring c : [ ℵ ] → ℵ . This is stronger than z p ( c ) ≤ λ , because it also implies that p [[ X i ] ] = λ for all i < λ . DVANCES ON STRONG COLORINGS OVER PARTITIONS 5 In [Eis13], Eisworth solved a longstanding open problem by proving that for anysingular cardinal λ , Pr ( λ + , λ + , λ, cf( λ )) implies Pr ( λ + , λ + , λ + , cf( λ )). In otherwords, it is possible to pump-up a strong coloring attaining λ many colors to oneattaining λ + many. It was asked in [CKS20] whether the same implication holdswith λ = ℵ — for strong colorings over partitions. In Section 6, we prove severalgeneral theorems in this vein. As a corollary we obtain a strong affirmative answerto [CKS20, Question 47] and a generalization (with a new proof) of Eisworth’spump-up theorem: Theorem D. For every infinite cardinal λ , every partition p : [ λ + ] → λ , andevery cardinal χ ≤ cf( λ ) , Pr ( λ + , λ + , λ, χ ) p ⇐⇒ Pr ( λ + , λ + , λ + , χ ) p . The proof of Theorem D brings Todorˇcevi´c’s method of walks on ordinals intothe study of strong colorings over partitions.In [She94, Lemma 4.5], Shelah proved that with suitable cardinal arithmeticassumptions, instances of Pr imply instances of Pr . [CKS20, Question 46] askswhether it is possible to pump-up Pr to Pr over a partition p at the level of ℵ .The following theorem provides a general affirmative answer. Theorem E. For a regular uncountable cardinal κ and cardinals µ, λ, χ, θ ≤ κ satisfying λ <χ < κ ≤ λ and λ <χ ≤ θ <χ = θ , for every partition p : [ κ ] → µ , Pr ( κ, κ, θ, χ ) p ⇐⇒ Pr ( κ, κ, θ, χ ) p . Notation and conventions. Throughout the paper, κ denotes a regular un-countable cardinal, χ, θ, µ, ν, ν ′ denote cardinals ≤ κ , and λ denotes an infinitecardinal < κ .For a set of ordinals a and an ordinal i < otp( a ), we write a ( i ) for the unique α ∈ a with otp( a ∩ α ) = i ; we also let acc( a ) := { α ∈ a | sup( a ∩ α ) = α > } . Forsets of ordinals a and b , we write a < b if α < β for all α ∈ a and β ∈ b . For acardinal χ and a set A , we write [ A ] χ := {B ⊆ A | |B| = χ } and [ A ] <χ := {B ⊆ A ||B| < χ } . This convention admits two refined exceptions: • For a set A which is either an ordinal or a collection of sets of ordinals, weinterpret [ A ] as { ( a, b ) ∈ A × A | a < b } ; • For an ordinal σ and a set of ordinals A , we write [ A ] σ for { B ⊆ A | otp( B ) = σ } .Our forcing convention is that p ≤ q means that p is a forcing condition whichextends the forcing condition q . We will follow the common convention of usingdotted free variables in forcing statement for forcing names and using undottedvariables for canonical names for sets from the ground model.2. Preliminaries This section collects several key definitions and a few useful facts. It is notintended for the reader to memorize all definitions below before reading on. Rather,the list of definitions here may serve as a glossary during reading. Lemma 2.9 aboutthe dominating number is needed only in Section 5. Definition 2.1. Let p : [ κ ] → µ be a partition and c : [ κ ] → θ be a coloring. Asubset A ⊆ κ is ( p, c ) -homogeneous iff there exists a function τ : µ → θ such that c ( α, β ) = τ ( p ( α, β )) for all ( α, β ) ∈ [ X ] . M. KOJMAN, A. RINOT, AND J. STEPR¯ANS The p -cochromatic number of c is the following cardinal: z p ( c ) := min { ζ ≤ κ | κ may be covered by ζ many ( p, c )-homogeneous sets } . Definition 2.2. Let p : [ κ ] → µ be a partition. A coloring c : [ κ ] → θ is said towitness • κ p [ κ ] θ iff for every A ⊆ κ of size κ there exists j < µ such that { c ( α, β ) | ( α, β ) ∈ [ A ] & p ( α, β ) = j } = θ ; • Pr ( κ, κ, θ, χ ) p iff for every pairwise disjoint family A ⊆ [ κ ] σ with |A| = κ and σ < χ , and every function τ : µ → θ there is ( a, b ) ∈ [ A ] such that c ( α, β ) = τ ( p ( α, β )) for all α ∈ a and β ∈ b ; • Pr ( κ, ν ⊛ κ (cid:30) ν ′ ⊛ κ , θ, χ ) p iff for every pairwise disjoint subfamilies A , B of [ κ ] σ with |A| = ν , |B| = κ and σ < χ there is A ′ ∈ [ A ] ν ′ such that for everyfunction τ : µ → θ , there are a ∈ A ′ and b ∈ B with a < b such that c ( α, β ) = τ ( p ( α, β )) for all α ∈ a and β ∈ b ; • Pr ( κ, κ, θ, χ ) p iff for every pairwise disjoint family A ⊆ [ κ ] σ with |A| = κ and σ < χ and every matrix ( τ i,j ) i,j<σ of functions from µ to θ there is apair ( a, b ) ∈ [ A ] such that c ( a ( i ) , b ( j )) = τ i,j ( p ( a ( i ) , b ( j ))) for all i, j < σ ; • Pr ( κ, ν ⊛ κ (cid:30) ν ′ ⊛ κ , θ, χ ) p iff for every pairwise disjoint subfamilies A , B of [ κ ] σ with |A| = ν , |B| = κ and σ < χ , there is A ′ ∈ [ A ] ν ′ such that for everymatrix ( τ i,j ) i,j<σ of functions from µ to θ , there are a ∈ A ′ and b ∈ B with a < b such that c ( a ( i ) , b ( j )) = τ i,j ( p ( a ( i ) , b ( j ))) for all i, j < σ. Convention . We write Pr i ( κ, ν ⊛ κ, θ, χ ) p for Pr i ( κ, ν ⊛ κ (cid:30) ν ⊛ κ , θ, χ ) p . We omitthe subscript p when p is constant (e.g, µ = 1).By [CKS20, Fact 5], for any partition p , Pr ( κ, κ, θ, p is equivalent to κ p [ κ ] θ .By [CKS20, Lemma 9], for any partition p : [ κ ] → µ and i < 2, Pr i ( κ, κ, θ µ , χ )implies Pr i ( κ, κ, θ, χ ) p . Definition 2.4 ([LR18]) . U( κ, κ, µ, χ ) asserts the existence of a function p : [ κ ] → µ such that for every σ < χ , every pairwise disjoint family A ⊆ [ κ ] σ of size κ , forevery δ < µ , there exists B ∈ [ A ] κ such that min( p [ a × b ]) > δ for all ( a, b ) ∈ [ B ] .By [LR18, Corollary 4.12], for every pair µ ≤ λ of infinite regular cardinals,U( λ + , λ + , µ, λ ) holds. Definition 2.5. For a partition p : [ κ ] → µ and a cardinal λ : • p has injective fibers iff for all α < α ′ < β , p ( α, β ) = p ( α ′ , β ); • p has <λ -to-one fibers iff for all β < κ and j < µ , |{ α < β | p ( α, β ) = j }| < λ ; • p has λ -almost-disjoint fibers iff for all β < β ′ < κ : |{ p ( α, β ) | α < β } ∩ { p ( α, β ′ ) | α < β }| < λ ; • p has λ -coherent fibers iff for all β < β ′ < κ : |{ α < β | p ( α, β ) = p ( α, β ′ ) }| < λ ; DVANCES ON STRONG COLORINGS OVER PARTITIONS 7 • p has λ -Cohen fibers iff for every injection g : a → µ with a ∈ [ κ ] <λ , thereare cofinally many β < κ such that g ( α ) = p ( α, β ) for all α ∈ a .At the level of ω , the following weak form of coherence is important: Definition 2.6. A partition p : [ ω ] → ω is ℓ ∞ -coherent iff for every ( β, β ′ ) ∈ [ ω ] , the set of integers { p ( α, β ) − p ( α, β ′ ) | α < β } is finite.An example of an ℓ ∞ -coherent partition which does not have ω -coherent fibersis the map ρ : [ ω ] → ω from the theory of walks on ordinals [Tod07, Defini-tion 2.4.1]. Proposition 2.7. Suppose that λ is an infinite regular cardinal. (1) There is a partition p : [ λ + ] → λ with injective and λ -coherent fibers; (2) For every cardinal χ such that λ <χ = λ , there is a partition p : [ λ + ] → λ with injective, λ -almost-disjoint and χ -Cohen fibers.Proof. (1) See, for instance, [Tod07, Lemma 6.25].(2) Assuming λ <χ = λ , fix an enumeration h g β | β < λ + i of all injections g withdom( g ) ∈ [ λ + ] <χ and Im( g ) ⊆ λ in which each such injection occurs cofinally often.For each β < λ + , let γ β := sup(Im( g β )) + 1.Fix a sequence ~Z = h Z β | β < λ + i of elements of [ λ ] λ such that, for all α < β <λ + , | Z α ∩ Z β | < λ . For all β < λ + and ι < λ , let Z β ( ι ) denote the unique ζ ∈ Z β to satisfy otp( Z β ∩ ζ ) = ι . For every β < λ + , fix an injection i β : β → λ . Then,define a partition p : [ λ + ] → λ via: p ( α, β ) := ( g β ( α ) if α ∈ dom( g β ); Z β ( γ β + i β ( α )) otherwise . A moment’s reflection makes it clear that p has injective and χ -Cohen fibers.To see that p is λ -almost-disjoint, fix an arbitrary pair ( β, β ′ ) ∈ [ λ + ] and con-sider the set A := { p ( α, β ) | α < β } ∩ { p ( α, β ′ ) | α < β } . Clearly, | A | ≤ | g β | + | g β ′ | + | Z β ∩ Z β ′ | < λ , as sought. (cid:3) The next proposition shows that if strong colorings exist over every partitionwith injective fibers, then they exist over every partition. Proposition 2.8. For every partition p : [ λ + ] → λ , there exists a correspondingpartition ¯ p : [ λ + ] → λ with injective fibers such that if any relation from Defini-tion 2.2 holds for ¯ p , then it also holds for p .Proof. Given p : [ λ + ] → λ , we define q : [ λ + ] → λ × λ as follows. Fix an arbitrarynonzero β < λ + . Fix a bijection i β : | β | ↔ β . Then, for every ǫ < | β | , let q ( i β ( ǫ ) , β ) := ( p ( i β ( ǫ ) , β ) , otp { ε < ǫ | p ( i β ( ε ) , β ) = p ( i β ( ǫ ) , β ) } ) . It is easy to check that, for all α < β < λ + : • q ( α, β ) = ( p ( α, β ) , ζ ) for some ζ < λ ; • q ( α ′ , β ) = q ( α, β ) for all α ′ < α .Finally, fix a bijection π : λ ↔ λ × λ and set ¯ p := π − ◦ q .Then, to any pattern τ ∈ λ θ , we define the corresponding pattern ¯ τ ∈ λ θ suchthat, for all η < λ , if π ( η ) = ( ξ, ζ ), then ¯ τ ( η ) = τ ( ξ ). (cid:3) M. KOJMAN, A. RINOT, AND J. STEPR¯ANS For many cardinal characteristics x of the continuum, the assertion “ x = ℵ ” maybe reformulated as a statement about the existence of a partition p : [ ω ] → ω withcertain properties. In Section 5 we shall need following reformulation of “ d = ℵ ”. Lemma 2.9. d = ℵ iff there exists a partition p : [ ω ] → ω with injective, ω -almost-disjoint and ω -Cohen fibers which satisfies the following:For every function h : ǫ → ω with ǫ < ω there exists γ < ω such that for every b ∈ [ ω \ γ ] < ℵ there exists ∆ ∈ [ ǫ ] < ℵ such that: • for all α ∈ ǫ \ ∆ and β ∈ b , h ( α ) < p ( α, β ) ; • p ↾ (( ǫ \ ∆) × b ) is injective.Proof. For the backwards implication, derive an ω -sized cofinal family { r β | ω ≤ β < ω } in ( ω ω, < ∗ ) by letting r β ( n ) := p ( n, β ).We turn now to the forward implication. By Proposition 2.7(1), fix q : [ ω ] → ω with injective and ω -coherent fibers. Fix an enumeration h g β | β < ω i of allinjections g with dom( g ) ∈ [ ω ] < ℵ and Im( g ) ⊆ ω in which each such injectionoccurs cofinally often. For each β < ω , let m β := sup(Im( g β )) + 1. Fix a bijection π : ω × ω ↔ ω . Derive a function ψ : ω → ω via ψ ( m ) := max { i < ω | ∃ j < ω ( π ( i, j ) ≤ m ) } . Using d = ℵ , it is easy to construct recursively a sequence ~d = h d β | β < ω i such that: • ~d is increasing and cofinal in ( ω ω, < ∗ ); • for every β < ω , min(Im( d β )) > ψ ( m β ).Finally, define a partition p : [ ω ] → ω via: p ( α, β ) := ( g β ( α ) if α ∈ dom( g β ); π ( d β ( q ( α, β )) , q ( α, β )) otherwise . Claim 2.9.1. Let α < β < ω . Then α ∈ dom( g β ) iff p ( α, β ) ∈ Im( g β ) .Proof. The forward implication is clear, so suppose that α / ∈ dom( g β ), and set j := q ( α, β ). By the choice of d β , i := d β ( j ) is greater than ψ ( m β ), and hence π ( i, j ) > m β > sup(Im( g β )). Altogether, p ( α, β ) = π ( i, j ) > sup(Im( g β )). (cid:3) As π is injective, q has injective fibers and each g β is injective, it follows that p has injective fibers. It is also clear that p has ω -Cohen fibers. Claim 2.9.2. p has ω -almost-disjoint fibers.Proof. Fix an arbitrary pair ( β, β ′ ) ∈ [ ω ] and consider the set A := { p ( α, β ) | α < β } ∩ { p ( α, β ′ ) | α < β } . Evidently, | A | ≤ m β + m β ′ + |{ n < ω | d β ( n ) = d β ′ ( n ) }| < ω . (cid:3) To see that p is as sought let us fix arbitrary ordinal ǫ < ω and function h : ǫ → ω . As q has ω -coherent fibers, for every β < ω above ǫ , the following setis finite A β := { α < ǫ | q ( α, ǫ ) = q ( α, β ) } . Define a real r : ω → ω via r ( n ) := ( ∀ α < ǫ ( q ( α, ǫ ) = n )); ψ ( h ( α )) if q ( α, ǫ ) = n. DVANCES ON STRONG COLORINGS OVER PARTITIONS 9 Find a large enough ordinal γ < ω such that ǫ < γ and r < ∗ d β for every β ∈ [ γ, ω ). Now, let b ∈ [ ω \ γ ] < ℵ be arbitrary. As q has injective fibers, forevery β ∈ [ γ, ω ), the following set is finite A β := { α < ǫ | r ( q ( α, β )) ≥ d β ( q ( α, β )) } . As ~d is < ∗ -increasing, we may find some m ∗ < ω such that, for all n ∈ [ m ∗ , ω )and ( β, β ′ ) ∈ [ b ] , d β ( n ) < d β ′ ( n ). Now, as q has injective fibers, it follows that thefollowing set is finite:∆ := [ β ∈ b ( A β ∪ A β ∪ dom( g β ) ∪ { α < ǫ | q ( α, β ) < m ∗ } ) . Claim 2.9.3. (1) for all α ∈ ǫ \ ∆ and β ∈ b , h ( α ) < p ( α, β ) ; (2) p ↾ (( ǫ \ ∆) × b ) is injective.Proof. (1) Let α ∈ ǫ \ ∆ and β ∈ b . Set n := q ( α, ǫ ). As α ∈ ǫ \ A β , q ( α, β ) = n ,so that ψ ( h ( α )) = r ( n ) = r ( q ( α, β )). As α ∈ ǫ \ A β , r ( q ( α, β )) < d β ( q ( α, β )).Altogether, ψ ( h ( α )) < d β ( q ( α, β )) and hence π ( d β ( q ( α, β )) , j ) > h ( α ) for all j < ω .In particular, since α / ∈ dom( g β ), p ( α, β ) = π ( d β ( q ( α, β )) , q ( α, β )) > h ( α ).(2) Fix ( α, β ) , ( α ′ , β ′ ) ∈ ( ǫ \ ∆) × b with p ( α, β ) = p ( α ′ , β ′ ). If β = β ′ , then since p has injective injective fibers, α = α ′ and we are done. So, suppose that β = β ′ , say, β < β ′ . Denote ( k, n ) := ( d β ( q ( α, β )) , q ( α, β )). As p ( α, β ) = p ( α ′ , β ′ ), α / ∈ dom( g β )and α ′ / ∈ dom( g β ′ ), it follows that ( d β ′ ( q ( α ′ , β ′ )) , q ( α ′ , β ′ )) = ( k, n ). In particular, d β ( n ) = d β ′ ( n ). As α ∈ ǫ \ ∆, we infer that n ≥ m ∗ , so d β ( n ) < d β ′ ( n ). This is acontradiction. (cid:3) This completes the proof. (cid:3) Strong colorings versus a nonmeager set Each of the following propositions is a consequence of CH :(L) There is an uncountable set of reals whose intersection with every meagerset is countable;(M) There is a nonmeager set of size ℵ ;(S) There is a sequence h f n | n < ω i of functions from from ω to ω such thatfor every uncountable I ⊆ ω , for all but finitely many n < ω , f n [ I ] = ω ;(EHM) There is a coloring c : [ ω ] → ω such that for every infinite A ⊆ ω andevery uncountable B ⊆ ω there exists α ∈ A such that c [ { α } × B ] = ω ;(G) There is a coloring d : [ ω ] → ω such that for every infinite pairwisedisjoint family A ⊆ [ ω ] < ℵ and every uncountable pairwise disjoint family B ⊆ [ ω ] < ℵ there exists a ∈ A such that for every δ < ω there exists some b ∈ B such that d [ a × b ] = { δ } .That CH implies (L) was shown by Mahlo and independently by Luzin around1913; such an uncountable set of reals is known as a Luzin set . That CH implies(S) was shown by Sierpi´nski in 1932, and may be found in his monograph [Sie34].That CH implies (EHM) was shown by Erd˝os, Hajnal and Milner in 1966 [EHM66].That CH implies (G) is easy to extract from Galvin’s work from 1980 [Gal80].Evidently, (L) = ⇒ (M) and (G) = ⇒ (EHM) = ⇒ (S). In 1980, Shelah [She80]established that (M) = ⇒ (L) (cf. [JS94]). In 1987, Todorˇcevi´c [Tod87, pp. 290–291]proved that (L) = ⇒ (S) ⇐⇒ (EHM). Recently, it was proved that (M) ⇐⇒ (S): thebackward implication is due to Miller [Mil14] and the forward implication is due to Guzm´an [Guz17]. Thus, (M) ⇐⇒ (S) ⇐⇒ (EHM). In this section, we expand thiscircle of equivalences, proving (M) ⇐⇒ (G), thereby improving Galvin’s theorem.Before stating our theorem, let us mention a few facts.Clause (1) of Theorem 3.1 below is proposition (M). Clause (2) is a syntacticweakening of proposition (S), but addressing a concern raised by Bagemihl andSprinkle [BS54], it was shown by Miller [Mil14] to be equivalent to it. Clause (3) isa higher-dimensional version of Clause (2). Clause (4) is a slight strengthening ofproposition (G): a coloring c : [ ω ] → ω × ω is gotten for which the map ( α, β ) δ iff ∃ ι [ c ( α, β ) = ( δ, ι )] witnesses Pr ( ℵ , ℵ ⊛ ℵ (cid:30) ⊛ ℵ , ℵ , ℵ ) of Definition 2.2 and, inaddition, has finite-to-one fibers. Clause (6) is proposition (EHM). The implication(7) = ⇒ (1) is due to Miller [Mil81] and the implication (1) = ⇒ (2) is due toGuzm´an [Guz17]. Theorem 3.1. The following are equivalent: (1) non( M ) = ℵ ; (2) There exists a sequence ~f = h f m | m < ω i of functions from ω to ω satisfying that for every cofinal subset B ⊆ ω there exists m < ω such that f m [ B ] = ω ; (3) There exists a sequence ~g = h g n | n < ω i of functions from ω to ω sat-isfying that for every uncountable pairwise disjoint subfamily B ⊆ [ ω ] < ℵ there are infinitely many n < ω such that for every γ < ω , for some b ∈ B , g n [ b ] = { γ } ; (4) There exists a coloring c : [ ω ] → ω × ω with finite-to-one fibers, such thatfor every • k < ω and an infinite pairwise disjoint subfamily A ⊆ [ ω ] k • l < ω and an uncountable pairwise disjoint subfamily B ⊆ [ ω ] l there exists a ∈ A such that for every δ < ω there are ι < ω and b ∈ B such that { α < β | c ( α, β ) = ( δ, ι ) } = a for every β ∈ b ;(5) For every ℓ ∞ -coherent partition p : [ ω ] → ω , there exists a correspondingcoloring d : [ ω ] → ω satisfying that for every • k < ω and an infinite pairwise disjoint subfamily A ⊆ [ ω ] k • l < ω and an uncountable pairwise disjoint subfamily B ⊆ [ ω ] l there exists a ∈ A such that for every matrix h τ n,m | n < k, m < l i offunctions from ω to ω there exists b ∈ B such that d ( a ( n ) , b ( m )) = τ n,m ( p ( a ( n ) , b ( m ))) for all n < k and m < l ;(6) There exists a coloring e : [ ω ] → ω such that for every infinite A ⊆ ω anduncountable B ⊆ ω , there is α ∈ A such that { e ( α, β ) | β ∈ B \ ( α +1) } = ω ; (7) There exists a subset X ⊆ ω ω of size ℵ with the property that for everyreal y : ω → ω , for some x ∈ X , x ∩ y is infinite.Proof. For the rest of the proof we fix a bijection π : ω ↔ ω × ω and, by theEngelking-Karlowicz theorem [EK65], we fix a sequence h h j | j < ω i of functionsfrom ω to ω such that for every set x ∈ [ ω ] < ℵ and a function h : x → ω thereexists j < ω such that h ⊆ h j .(1) = ⇒ (2): This is Proposition 2.2 of [Guz17].(2) = ⇒ (3): Let ~f witness Clause (2). For every β < ω fix a surjection e β : ω → β + 1. Define a sequence ~g = h g n | n < ω i of functions from ω to ω , as DVANCES ON STRONG COLORINGS OVER PARTITIONS 11 follows. Given n < ω , let ( m, j ) := π ( n ) and for every β < ω set g n ( β ) := f m ( e β ( h j ( β ))) . To see that ~g witnesses Clause (3), fix an arbitrary uncountable pairwise disjointsubfamily B ⊆ [ ω ] < ℵ and some k < ω . We shall find an integer n > k such that,for every γ < ω there is some b ∈ B such that g n [ b ] = { γ } .For every b ∈ B , define a function h b : b → ω via: h b ( β ) := min { i < ω | e β ( i ) = min( b ) } . Fix j ′ < ω for which B ′ := { b ∈ B | h b ⊆ h j ′ } is uncountable, and then let m ′ := max( { } ∪ { m < ω | π − ( m, j ′ ) ≤ k } ) . Evidently, B := { min( b ) | b ∈ B ′ } is uncountable. Next, for every i ≤ m ′ such that B i has already been defined, we do the following: ◮ If Z i := { β ∈ B i | f i ( β ) = 0 } is uncountable, then let B i +1 := Z i ; ◮ Otherwise, let B i +1 := B i \ Z i .In either case, B i +1 ⊆ B i is uncountable with f i [ B i +1 ] = ω .Finally, as B m ′ +1 is uncountable, let us pick, by the choice of ~f , an integer m < ω such that f m [ B m ′ +1 ] = ω .For all i ≤ m ′ , f i [ B m ′ +1 ] ⊆ f i [ B i +1 ] ( ω , so m > m ′ . In particular, n := π ( m, j ′ ) is larger than k . To see that n is as sought, let γ ∈ ω = f m [ B m ′ +1 ] bearbitrary. Pick β ′ ∈ B m ′ +1 with f m ( β ′ ) = γ . As β ′ ∈ B m ′ +1 ⊆ B , let us pick b ∈ B such that h b ⊆ h j ′ and min( b ) = β ′ . Let β ∈ b be arbitrary. Then g n ( β ) = f m ( e β ( h j ′ ( β ))) = f m ( e β ( h b ( β ))) = f m ( β ′ ) = γ. So g n [ b ] = { γ } , as required.(3) = ⇒ (4): The proof here is inspired by Miller’s proof of [Mil14, Propo-sition 4]. Define an eventually increasing sequence of integers h m n | n < ω i byrecursion, setting m := 1, and m n +1 := n ! · ( P i ≤ n m i ) for every n < ω . For every n < ω , let Φ n := S { x ( ω × ω ) | x ⊆ ω , | x | = m n } . Evidently, | Φ n | = ω , so wemay fix an injective enumeration h φ γn | γ < ω i of Φ n .Let ~g witness Clause (3). Define a coloring d : [ ω ] → ( ω × ω ) × ω by lettingfor all α < β < ω : d ( α, β ) := ( (( α, , 0) if α / ∈ S i<ω dom( φ g i ( β ) i );( φ g n ( β ) n ( α ) , n + 1) if n = min { i < ω | α ∈ dom( φ g i ( β ) i ) } . Finally, define a coloring c : [ ω ] → ω × ω by letting c ( α, β ) := ( γ, π ( ι, n )) iff d ( α, β ) = (( γ, ι ) , n ). It is clear that d has finite-to-one fibers, and hence so does c .To see that c witnesses Clause (4), fix positive integers k, l along with A , B suchthat: • A is an infinite pairwise disjoint subfamily of [ ω ] k , • B is an uncountable pairwise disjoint subfamily of [ ω ] l , and • a < b for all a ∈ A and b ∈ B .By the choice of ~g , let us fix an integer n > max { k, l } such that, for every γ < ω ,for some b ∈ B , g n +1 [ b ] = { γ } . As m n +1 is divisible by k , we now fix an injectivesequence h a ι | ι < m n +1 k i consisting of elements of A . Claim 3.1.1. There exists ι < m n +1 k such that, for every δ < ω , there is b ∈ B ,such that, for every β ∈ b : { α < β | c ( α, β ) = ( δ, π ( ι, n + 2)) } = a ι . Proof. Suppose not. Then, for every ι < m n +1 k , we may find some δ ι < ω suchthat for all b ∈ B , for some β ∈ b , { α < β | d ( α, β ) = (( δ ι , ι ) , n + 2) } 6 = a ι . Define a function φ : U { a ι | ι < m n +1 k } → ω × ω by letting φ ( α ) := ( δ ι , ι ) iff α ∈ a ι .As | U { a ι | ι < m n +1 k }| = m n +1 k · k = m n +1 , we infer that φ ∈ Φ n +1 , so we may fix γ < ω such that φ = φ γn +1 . Now, pick b ∈ B with g n +1 [ b ] = { γ } .For every i ≤ n and β ∈ b , let x βi := dom( φ g i ( β ) i ), so that | x βi | = m i . Next, set x := S { x βi | i ≤ n, β ∈ b } . As | b | = l , we infer that | x | ≤ l · P i ≤ n m i . Thus k · | x | ≤ k · l · X i ≤ n m i < n ! · X i ≤ n m i = m n +1 . In particular, | x | < m n +1 k , so we may fix ι < m n +1 k such that a ι ∩ x = ∅ .Let β ∈ b be arbitrary. Consider the set A := { α < β | d ( α, β ) = (( δ ι , ι ) , n + 2) } . As g n +1 ( β ) = γ , we infer that φ g n +1 ( β ) n +1 = φ , so, by the definition of d : A ⊆ { α < β | φ g n +1 ( β ) n +1 ( α ) = ( δ ι , ι ) } ⊆ { α < β | φ ( β )( α ) = ( δ ι , ι ) } = a ι . On the other hand, for every α ∈ a ι ⊆ dom( φ g n +1 ( β ) n +1 ), as α / ∈ x , it follows thatmin { i < ω | α ∈ dom( φ g i ( β ) i ) } = n + 1, and hence d ( α, β ) = ( φ g n +1 ( β ) n +1 ( α ) , n + 2) = ( φ ( α ) , n + 2) = (( δ ι , ι ) , n + 2) , so that α ∈ A i . Altogether, A = a ι , contradicting the choice of δ ι . (cid:3) (4) = ⇒ (5): Fix c witnessing Clause (4). Let h η γ | γ < ω i be some injectiveenumeration of S { k × l × t ω | k, l, t < ω } and let h ( i δ , j δ , γ δ ) | δ < ω i be someinjective enumeration of ω × ω × ω ,Now, given any ℓ ∞ -coherent partition p : [ ω ] → ω , define a coloring d : [ ω ] → ω as follows. Given ( α, β ) ∈ [ ω ] , let ( δ, ι ) := c ( α, β ) and then set d ( α, β ) := ( η γ δ ( h i δ ( α ) , h j δ ( β ) , p ( α, β )) if ( h i δ ( α ) , h j δ ( β ) , p ( α, β )) ∈ dom( η γ δ )0 otherwise . To see that d witnesses Clause (5), fix k, l, A , B and ǫ < ω such that: • A is an infinite pairwise disjoint subfamily of [ ω ] k , • B is an uncountable pairwise disjoint subfamily of [ ω ] l and • max( a ) < ǫ ≤ min( b ) for all a ∈ A and all b ∈ B .For every x ∈ A ∪ B , define a function h x : x → ω via: h x ( β ) := otp( x ∩ β ) . Now pick j ′ < ω for which B ′ := { b ∈ B | h b ⊆ h j ′ } is uncountable. As p is ℓ ∞ -coherent, we may shrink B ′ further and assume the existence of some q < ω such that, for all b ∈ B ′ : {| p ( α, ǫ ) − p ( α, β ) | | β ∈ b } ⊆ q. DVANCES ON STRONG COLORINGS OVER PARTITIONS 13 Now, as |A| = ℵ and |B ′ | = ℵ , by the choice of c , we may fix a ∈ A such that,for every δ < ω , there are b ∈ B ′ and ι < ω such that c [ a × b ] = { ( δ, ι ) } . Claim 3.1.2. Let h τ n,m | n < k, m < l i be a matrix of functions from ω to ω .Then there exists b ∈ B ′ satisfying that, for all n < k and m < l , d ( a ( n ) , b ( m )) = τ n,m ( p ( a ( n ) , b ( m ))) . Proof. Fix i ′ < ω such that h a ⊆ h i ′ . Let t := max { p ( α, ǫ ) + q | α ∈ a } . Define afunction η : k × l × t → ω via: η ( n, m, s ) := τ n,m ( s ) . Let δ < ω be such that ( i δ , j δ , η γ δ ) = ( i ′ , j ′ , η ). Pick b ∈ B ′ and ι < ω such that c [ a × b ] = { ( δ, ι ) } . Now, given n < k and m < l , we have c ( a ( n ) , b ( m )) = ( δ, ι ), p ( a ( n ) , b ( m )) < p ( a ( n ) , ǫ ) + q ≤ t , so that d ( a ( n ) , b ( m )) = η γ δ ( h i δ ( a ( n )) , h j δ ( b ( m )) , p ( a ( n ) , b ( m )))= η ( h a ( a ( n )) , h b ( b ( m )) , p ( a ( n ) , b ( m )))= η ( n, m, p ( a ( n ) , b ( m )))= τ n,m ( p ( a ( n ) , b ( m ))) , as sought. (cid:3) (5) = ⇒ (6): Let d witness Clause (5) with respect to the constant partition p : [ ω ] → 1. Define a function e : [ ω ] → ω via e ( α, β ) := ( d ( α, β ) if d ( α, β ) < ω ;0 otherwise . Clearly, e witnesses (6).(6) = ⇒ (7): Let e witness Clause (6). Define X = { x β | β < ω } , as follows.For every β < ω , define a function x β : ω → ω via x β ( n ) := e ( n, β ). Towardsa contradiction, suppose that y : ω → ω is a counterexample. It follows thatthere exists a large enough n < ω for which B := { β < ω | dom( x β ∩ y ) ⊆ n } is uncountable. By the choice of e , we may now fix an integer α > n such that { e ( α, β ) | β ∈ B \ ( α + 1) } = ω . In particular, we may find β ∈ B such that e ( α, β ) = y ( α ). Altogether, x β ( α ) = e ( α, β ) = y ( α ) contradicting the fact that β ∈ B and α > n .(7) = ⇒ (1): By Theorem 1.3 of [Mil81]. (cid:3) Thus, we arrived at Theorem A: Corollary 3.2. non ( M ) = ℵ iff Pr ( ℵ , ℵ ⊛ ℵ (cid:30) ⊛ ℵ , ℵ , ℵ ) p holds for every ℓ ∞ -coherent partition p : [ ω ] → ω . (cid:3) Corollary 3.3. In the following, (1) = ⇒ (2) = ⇒ (3) and none of the implica-tions is revertible. (1) Pr ( ℵ , ℵ ⊛ ℵ (cid:30) ⊛ ℵ , ℵ , ℵ ) ; (2) Pr ( ℵ , ℵ , ℵ , ℵ ) ; (3) Pr ( ℵ , ℵ , ℵ , n ) for all n < ω .Proof. To see that (2) does not imply (1), recall that non( M ) > ℵ = b is consistent(e.g., after adding ℵ random reals to a model of CH ) and that Todorˇcevi´c [Tod88]proved that Clause (2) is a consequence of b = ℵ . To see that (3) does not imply (2) recall that Clause (2) is refuted by MA ℵ , andthat Peng and Wu [PW18] proved Clause (3) in ZFC . (cid:3) Theorem 3.4. If C ℵ is the partial order for adding ℵ Cohen reals then for everysequence p = h p δ : δ < ω i of partitions p δ : [ ω ] → ω in the forcing extension by C ℵ , (cid:13) C ℵ Pr ( ℵ , ℵ ⊛ ℵ (cid:30) ⊛ ℵ , ℵ , ℵ ) p . Proof. Let C α be the partial order of finite partial functions from [ α ] to ω . Let V be a model of set theory and let G ⊆ C ω be generic over V . Then S G : [ ω ] → ω .Now suppose that p = h p δ | δ < ω i is an arbitrary sequence of partitions p δ : [ ω ] → ω in V [ G ]. As there is some α ∈ ω such that p ∈ V [ G ∩ C α ], it may beassumed that p ∈ V . Let c = S G ↾ [ ω ] . So c : [ ω ] → ω . In V , for every infiniteordinal α < ω fix a bijection e α : ω ↔ α . Define a coloring f : [ ω ] → ω by f ( α, β ) = e β ( c ( α, β )) , for β ≥ ω and as 0 otherwise.To see that f witnesses Pr ( ℵ , ℵ ⊛ ℵ (cid:30) ⊛ ℵ , ℵ , ℵ ) p suppose that there is some δ < ω for which f fails to witness Pr ( ℵ , ℵ ⊛ ℵ (cid:30) ⊛ ℵ , ℵ , ℵ ) p . This means thatfor some n ∈ ω there are in V G pairwise disjoint A ⊆ [ ω ] n and B ∈ [ ω ] n suchthat for every a ∈ A there is some W ( a ) : ω → M n × n ( ω ) such that for all b ∈ B such that max a < min b it holds that f (( a ( i ) , b ( j )) = W ( a ) i,j ( p ( a ( i ) , b ( j )) for some i, j < n . Let ˙ A and ˙ W be countable names for A and W and let ˙ B be a name for B . Let r ∈ G decide δ and force r (cid:13) ( ∀ a ∈ ˙ A )( ∀ b ∈ ˙ B ) a < b = ⇒ ( ∃ i, j < n )( f ( a ( i ) , b ( j ))) = ˙ W ( a ) i,j ( p ( a ( i ) , b ( j )))Let M be a countable elementary submodel of h H ( θ ) , ˙ A , ˙ B , ˙ W , r i , for some suf-ficiently large regular θ .Fix an extension r ′ ∈ G of r and and a member b ∈ B with min b > sup( M ∩ ω )such that r ′ (cid:13) b ∈ ˙ B . Let r = r ′ ∩ M . Inside M extend r to r such that r (cid:13) a ∈ ˙ A for some a ∈ A which is disjoint to S dom( r ′ ) and r decides W ( a )( p ( a ( i ) , b ( j )) forall i, j < n . Thus, { a ( i ) , b ( j ) } / ∈ dom( r ′ ∪ r ) for all i, j < n . Let r ∗ = r ′ ∪ r ∪ (cid:8) h{ a ( i ) , b ( j ) } , e − b ( j ) ( W ( a ) i,j ( p ( a ( i ) , b ( j ))) i | i, j < n (cid:9) . Since r ∗ extends r and f ( a ( i ) , b ( j )) = e b ( j ) ( c ( a ( i ) , b ( j ()) = W ( a )( p ( a ( i )) , b ( j )))for all i, j < n , this is a contradiction to the choice of r . (cid:3) Partitions and forcing axioms A notion of forcing Q has Knaster’s Property (Property K) iff for every uncount-able set A of condition in Q , there is an uncountable B ⊆ A such that any twoelements of B are compatible. Definition 4.1. MA ℵ ( K ) asserts that for every notion of forcing Q having Prop-erty K, for every sequence h D β | β < ω i of dense subsets of Q , there is a filter G over Q that intersects each of the D β ’s.A notion of forcing Q = ( Q, ≤ ) is well-met iff every pair q , q of compatibleconditions has a greatest lower bound, i.e., an r ≤ q , q such that for any condition s , if s ≤ q , q then s ≤ r . The notion of forcing Q is said to satisfy the λ + -stationarychain condition ( λ + -stationary-cc, for short) iff for every sequence h q δ | δ < λ + i of DVANCES ON STRONG COLORINGS OVER PARTITIONS 15 conditions in Q there is a club D ⊆ λ + and a regressive map h : D ∩ E λ + cf( λ ) → λ + such that for all γ, δ ∈ dom( h ), if h ( γ ) = h ( δ ) then q γ and q δ are compatible. Definition 4.2 (Generalized Martin’s Axiom) . GMA λ + asserts that for every no-tion of forcing Q = ( Q, ≤ ) of size < λ which satisfies the following conditions:(a) Q is well-met;(b) For all σ < λ , every ≤ -decreasing sequence of condition h q i | i < σ i in Q admits a greatest lower bound;(c) Q satisfy the λ + -stationary-cc,for every sequence h D β | β < λ + i of dense subsets of Q there is a filter G over Q that meets each of the D β ’s.By Fodor’s lemma, any poset satisfying the ω -stationary-cc has Property K .Thus, MA ℵ = ⇒ MA ℵ ( K ) = ⇒ GMA ℵ . Fact 4.3 (Shelah, [She78]) . Suppose the GCH holds. Then for any prescribed regularcardinal λ there is a cofinality-preserving forcing extension in which λ <λ = λ and GMA λ + holds.Remark . The conjunction of λ <λ = λ and GMA λ + implies that 2 λ > λ + .Otherwise, fix an enumeration h f β | β < λ + i of λ λ and appeal to GMA λ + with Q := Add( λ, 1) and D β := { q ∈ <λ λ | q * f β } for each β < λ + .Together with Proposition 2.7(2), the next result is Clause (1) of Theorem B. Theorem 4.5. Suppose λ <λ = λ and GMA λ + holds. Let p : [ λ + ] → λ be anypartition with injective and λ -almost-disjoint fibers.For every coloring c : [ λ + ] → λ , there exists a decomposition λ + = U i<λ X i such that for all i < λ : • X i is ( p, c ) -homogeneous (recall Definition 2.1); • if p has in addition λ -Cohen fibers, then | X i | = λ + and p [[ X i ] ] = λ .In particular, for every coloring c : [ λ + ] → λ , z p ( c ) ≤ λ .Proof. Fix an arbitrary coloring c : [ λ + ] → λ . Define a notion of forcing Q =( Q, ⊇ ), where Q consists of all functions f : a → λ such that:(1) a ∈ [ λ + ] <λ ;(2) for all i, j < λ , the setΓ fi,j := { c ( α, β ) | ( α, β ) ∈ [ a ] , f ( α ) = i = f ( β ) , p ( α, β ) = j } contains at most one element.It will be shown that Q satisfies the requirement of Definition 4.2. Claim 4.5.1. Let F be a centered family of conditions of size < λ . Then S F is acondition.Proof. Suppose not. Denote f := S F and a := dom( f ). As a ∈ [ λ + ] <λ this mustmean that there are i, j < λ for which the set Γ fi,j has more than two elements.This means that we may pick ( α , β ) , ( α , β ) ∈ [ a ] such that: • f ( α ) = f ( α ) = i = f ( β ) = f ( β ); • p ( α , β ) = j = p ( α , β ); • c ( α , β ) = c ( α , β ). Fix f , f , f , f ∈ F such that α ∈ dom( f ), α ∈ dom( f ), β ∈ dom( f )and β ∈ dom( f ). Since F is centered, there exists a condition q such that f ∪ f ∪ f ∪ f ⊆ q . A moment’s reflection makes it clear that q ( α ) = q ( α ) = i = q ( β ) = q ( β ), so since p ( α , β ) = j = p ( α , β ) and since q is legitimatecondition, this must mean that c ( α , β ) = c ( α , β ). This is a contradiction. (cid:3) Claim 4.5.2. For every β < λ + , D β := { f ∈ Q | β ∈ dom( f ) } is dense in Q .Proof. Let β < λ + . Given any condition f such that β / ∈ dom( f ), we get that f ′ := f ∪ { ( β, sup(Im( f )) + 1) } is an extension of f in D β . (cid:3) It thus follows that if G is a filter over Q that meets each of the D β ’s, then g := S G is a function from λ + to λ such that for every i < λ , if we let X i := { α < λ + | g ( α ) = i } , then |{ c ( α, β ) | ( α, β ) ∈ [ X i ] & p ( α, β ) = j }| ≤ j < λ . Thus, z p ( c ) ≤ λ holds in the forcing extenstion by Q . Claim 4.5.3. If p has λ -Cohen fibers, then, for all ǫ < λ + and i, j < λ : (1) D ǫ := { f ∈ Q | ∃ β ∈ dom( f ) ( β ≥ ǫ & f ( β ) = i ) } is dense; (2) D i,j := { f ∈ Q | ∃ ( α, β ) ∈ [dom( f )] , f ( α ) = i = f ( β ) & p ( α, β ) = j } isdense.Proof. Suppose that p has λ -Cohen fibers.(1) Given ǫ < λ + and a condition f : a → λ , fix a large enough η < λ + suchthat { p ( α, β ) | ( α, β ) ∈ [ a ] } ⊆ η , and then define an injection g : a → λ via g ( α ) := η + otp( a ∩ α ) . Now, as p has λ -Cohen fibers, we may find some β < λ + with a ∪ ǫ ⊆ β such that p ( α, β ) = g ( α ) for all α ∈ a . It is clear that f ′ := f ∪ { ( β, i ) } is an extension of f lying in D ǫ .(2) Given i, j < λ + and a condition f : a → λ , we do the following. First, byClause (1), we may assume the existence of some α ∗ ∈ a such that f ( α ∗ ) = i . Ofcourse, if f ∈ D i,j , then we are done, thus, hereafter, assume that f / ∈ D i,j .Fix a large enough η ∈ λ + \ ( j + 1) such that { p ( α, β ) | ( α, β ) ∈ [ a ] } ⊆ η . Definean injection g : a → λ via g ( α ) := ( j if α = α ∗ ; η + otp( a ∩ α ) otherwise . Now, as p has λ -Cohen fibers, we may find some β < λ + with a ⊆ β suchthat p ( α, β ) = g ( α ) for all α ∈ a . As f D i,j , it immediately follows that f ′ := f ∪ { ( β, i ) } is a legitimate condition lying in D i,j . So we are done. (cid:3) Clearly, Q has size no more than | [ λ + ] <λ | = λ + < λ . In addition, by Claim 4.5.1,Clauses (a) and (b) of Definition 4.2 hold true. Thus, to complete the proof, we areleft with addressing Clause (c) of Definition 4.2. To this end, assume we are givena sequence h f δ | δ < λ + i of conditions in Q ; we need to find a club D ⊆ λ + and aregressive map h : D ∩ E λ + λ → λ + such that for all γ, δ ∈ D ∩ E λ + λ , if h ( γ ) = h ( δ )then f γ and f δ are compatible.Consider the club C := { δ < λ + | ∀ γ < δ (sup(dom( f γ )) < δ ) } . Fix an injectiveenumeration h ( ψ τ , ϕ τ , ξ τ , µ τ , ǫ τ ) | τ < λ + i of [ λ ] <λ × [ λ + × λ ] <λ × λ × λ × λ + , andthen consider the subclub: D := { δ ∈ C | { ( ψ τ , ϕ τ , ξ τ , µ τ , ǫ τ ) | τ < δ } = [ λ ] <λ × [ δ × λ ] <λ × λ × λ × δ } . DVANCES ON STRONG COLORINGS OVER PARTITIONS 17 We define the function h : D ∩ E λ + λ → λ + as follows. Given δ ∈ D ∩ E λ + λ , let h ( δ ) := τ for the least τ < δ which satisfies all of the following:(a) (cid:8) ( i, j, c ( α, β )) | i, j < λ, ( α, β ) ∈ [dom( f δ )] , f δ ( α ) = i = f δ ( β ) , p ( α, β ) = j (cid:9) = ψ τ ;(b) f δ ↾ δ = ϕ τ ;(c) S {{ p ( α, β ) | α < β } ∩ { p ( α, β ′ ) | α < β } | ( β, β ′ ) ∈ [dom( f δ )] } ⊆ ξ τ ;(d) { p ( α, β ) | ( α, β ) ∈ [dom( f δ )] } ⊆ µ τ ;(e) { α < δ | ∃ β ∈ dom( f δ ) \ δ [ p ( α, β ) ≤ max { ξ τ , µ τ } ] } ⊆ ǫ τ . Claim 4.5.4. h is well-defined.Proof. Let δ ∈ D ∩ E λ + λ . First we make note of the following: • The corresponding set of Clause (a) is a subset of λ of size < λ ; • The corresponding set of Clause (b) is a subset of δ × λ of size < λ ; • As | dom( f δ ) | < λ , the fact that p has λ -almost-disjoint fibers ensures thatan ordinal ξ τ < λ as in Clause (c) exists; • As | dom( f δ ) | < λ , an ordinal µ τ < λ as in Clause (d) does exist; • As | dom( f δ ) | < λ = cf( δ ), the fact that p has injective fibers ensures thatan ordinal ǫ τ < λ as in Clause (e) exists.So, since δ ∈ D , a τ < δ for which Clause (a)–(e) are satisfied does exist. (cid:3) To see that h is as sought, fix a pair γ < δ of ordinals in D ∩ E λ + λ such that h ( γ ) = h ( δ ), say, both are equal to τ . As δ ∈ C , Clause (b) implies that f := f γ ∪ f δ is afunction. To see that f ∈ Q , it suffices to verify Clause (2) above with a := dom( f ).Towards a contradiction, suppose that there i, j < λ and ( α , β ) , ( α , β ) ∈ [ a ] such that: • f ( α ) = f ( α ) = i = f ( β ) = f ( β ); • p ( α , β ) = j = p ( α , β ); • c ( α , β ) = c ( α , β ).Denote a γ := dom( f γ ) and a δ := dom( f δ ). As h ( γ ) = τ = h ( δ ), Clause (a)implies that it cannot be the case that { ( α , β ) , ( α , β ) } ⊆ [ a γ ] ∪ [ a δ ] . So,without loss of generality, assume that ( α , β ) / ∈ [ a γ ] ∪ [ a δ ] . By Clause (b), inparticular, a γ ∩ γ = a δ ∩ δ , and so, since ( α , β ) / ∈ [ a γ ] ∪ [ a δ ] , it must be thecase that α ≥ γ and β ≥ δ . If α ≥ δ , then since δ ∈ C , we would get that( α , β ) ∈ [ a δ ] , which is not the case. Altogether, γ ≤ α < δ ≤ β . In particular, α ∈ ( a γ \ γ ) and β ∈ ( a δ \ δ ).By Clause (e), ǫ τ < γ , and hence α > ǫ τ . It thus follows from Clause (e) that p ( α , β ) > max { ξ τ , µ τ } . So p ( α , β ) = j = p ( α , β ) > µ τ . Recalling Clause (d),this means that ( α , β ) / ∈ [ a γ ] ∪ [ a δ ] . Hence, the same analysis we had for ( α , β )is valid also for ( α , β ). In particular, γ ≤ α < δ ≤ β (so that { β , β } ⊆ a δ \ δ )and p ( α , β ) > ξ τ . By Clause (c) for γ , ξ τ < γ < α , α and then by Clause (c) for δ we infer that if β = β , then p ( α , β ) = p ( α , β ). It thus follows that β = β .As p has has injective fibers it follows that α = α , contradicting the fact that c ( α , β ) = c ( α , β ). (cid:3) The next result answers two questions from [CKS20]; it answers Question 48 inthe negative and Question 49 in the affirmative. Corollary 4.6. Assume MA ℵ ( K ) . Then there exists a partition p : [ ω ] → ω suchthat for every coloring c : [ ω ] → ω there exists a decomposition ℵ = U i<ω X i such that, for every i < ω : • X i is uncountable; • X i is p -omnichromatic, i.e., p [[ X i ] ] = ω ; • X i is ( p, c ) -homogeneous, i.e., for every j < ω , c ↾ { ( α, β ) ∈ [ X i ] | p ( α, β ) = j } is constant . Proof. By Theorem 4.5 and Proposition 2.7(2). (cid:3) The next theorem applies to a broader class of partitions, in return for admittinga finite number of colors rather than a constant. It also implies Clause (2) ofTheorem B. Theorem 4.7. Suppose MA ℵ ( K ) holds and that p is any partition witnessing U( ω , ω , ω, ω ) . For every coloring c : [ ω ] → ω , there is a decomposition ω = U i<ω X i such that, for all i, j < ω , { c ( α, β ) | ( α, β ) ∈ [ X i ] & p ( α, β ) = j } is finite . Proof. Define a notion of forcing Q consisting of conditions q = ( m q , f q , g q ) asfollows:(1) m q < ω ;(2) f q : m q × m q → ω is a function;(3) g q : a q → m q is a function, with a q ∈ [ ω ] < ℵ ;(4) for all ( α, β ) ∈ [ a q ] , p ( α, β ) < m q ;(5) for all ( α, β ) ∈ [ a q ] , if g q ( α ) = g q ( β ), then c ( α, β ) < f q ( g q ( α ) , p ( α, β )).A condition q extends a condition ¯ q iff m q ≥ m ¯ q , f q ⊇ f ¯ q and g q ⊇ g ¯ q . Claim 4.7.1. For every β < ω , D β := { q | β ∈ a q } is dense in Q Proof. Let β < ω . Given any condition ( m, f, g ) in Q such that β / ∈ dom( g ), let m ′ := max { m, p ( α, β ) | α ∈ dom( g ) } and define a condition q = ( m q , f q , g q ), by letting m q := m ′ + 1, letting f q : m q × m q → ω be an arbitrary function extending f , and letting g q := g ∪ { ( β, m ′ ) } . (cid:3) It thus follows that if G is a filter over Q such that G ∩ D β = ∅ for all β < ω , thenby letting f := S { f q | q ∈ G } and g := S { g q | q ∈ G } , we get that dom( f ) = ω × ω ,dom( g ) = ω , and for every i, j < ω , if we let X i := { β < ω | g ( β ) = i } , then { c ( α, β ) | ( α, β ) ∈ [ X i ] & p ( α, β ) = j } ⊆ f ( i, j ).Now, let us verify that Q has Property K . To this end, suppose that A is anuncountable family of conditions in Q . By the pigeonhole principle, we may assumethe existence of an integer m and a function f such that, for all q ∈ A , m q = m and f q = f . By the ∆-system lemma, we may also assume that { a q | q ∈ A } formsa head-tail-tail ∆-system with some root r . By shrinking further, we may assumethat q g q ↾ r is constant over A . For any q ∈ A , denote a ′ q := a q \ r , so that A := { a ′ q | q ∈ A } forms an uncountable pairwise disjoint family. Next, by thechoice of the partition p , we fix an uncountable B ⊆ A with the property that forall ¯ q, q ∈ B , if max( a ′ ¯ q ) < min( a ′ q ), then min( p [ a ′ ¯ q × a ′ q ]) > m .To see that { a q | q ∈ B } is directed, fix two conditions ¯ q = q in B . By thehead-tail-tail pattern of our ∆-system, we may assume that max( a ′ ¯ q ) < min( a ′ q ). DVANCES ON STRONG COLORINGS OVER PARTITIONS 19 Set a ∗ := a ¯ q ∪ a q , g ∗ := g ¯ q ∪ g q and m ∗ := max( p [ a ∗ ] ) + 1. Claim 4.7.2. For every ( α, β ) ∈ [ a ∗ ] \ ([ a ¯ q ] ∪ [ a q ] ) , ( α, β ) ∈ a ′ ¯ q × a ′ q .Proof. Let ( α, β ) ∈ [ a ∗ ] \ ([ a ¯ q ] ∪ [ a q ] ). As r = a ¯ q ∩ a q , we infer that { α, β } ∩ r = ∅ .So { α, β } ∩ a ′ ¯ q and { α, β } ∩ a ′ q are singletons. Since α < β and max( a ′ ¯ q ) < min( a ′ q ),it altogether follows that ( α, β ) ∈ a ′ ¯ q × a ′ q . (cid:3) Fix any function f ∗ : m ∗ × m ∗ → ω extending f by letting, for all ( i, j ) ∈ ( m ∗ × m ∗ ) \ ( m × m ), f ∗ ( i, j ) := max { c ( α, β ) | ( α, β ) ∈ [ a ∗ ] , g ( α ) = i = g ( β ) & p ( α, β ) = j ) } + 1 . Looking at Clauses (1)–(5) above, it is clear that for q ∗ := ( m ∗ , f ∗ , g ∗ ) to be acondition in Q it suffices to verify the following claim. Claim 4.7.3. Let ( α, β ) ∈ [ a ∗ ] with g ∗ ( α ) = g ∗ ( β ) . Then c ( α, β ) < f ∗ ( g ∗ ( α ) , p ( α, β )) . Proof. Denote i := g ∗ ( α ) and j := p ( α, β ). We shall show that c ( α, β ) < f ∗ ( i, j ).Of course, if ( α, β ) ∈ [ a ¯ q ] , then g ¯ q ( α ) = i < m , p ( α, β ) < m and c ( α, β ) = f ¯ q ( i, j ) = f ∗ ( i, j ). Likewise, if ( α, β ) ∈ [ a q ] , then c ( α, β ) = f q ( i, j ) = f ∗ ( i, j ).Next, assume that ( α, β ) / ∈ [ a ¯ q ] ∪ [ a q ] . So, by Claim 4.7.2, ( α, β ) ∈ a ′ ¯ q × a ′ q . As j ≥ min( p [ a ′ ¯ q × a ′ q ]) > m > max( p [[ a ¯ q ] ∪ [ a q ] ]), we infer that ( i, j ) ∈ ( m ∗ × m ∗ ) \ ( m × m ) and hence the definition of f ∗ ( i, j ) makes it clear that c ( α, β ) < f ∗ ( i, j ),as sought. (cid:3) So q ∗ is a legitimate condition witnessing that ¯ q and q are compatible. Thus, wehave demonstrated that Q indeed satisfies Property K . (cid:3) We now present two ZFC results which show that the preceding is optimal. To seehow the first result connects to Theorem 4.7 note that any partition p : [ ω ] → ω with injective (or just <ω -to-one) fibers witnesses U( ω , ω , ω, ω ). Theorem 4.8. There exist a partition p : [ ω ] → ω with injective fibers and acoloring c : [ ω ] → ω such that, for every k < ω , and every X ⊆ ω with otp( X ) = ω + k , there exists j < ω such that |{ c ( α, β ) | ( α, β ) ∈ [ X ] & p ( α, β ) = j }| ≥ k .Proof. By Proposition 2.7, let us fix a partition p : [ ω ] → ω with injective and ω -coherent fibers, and a coloring c : [ ω ] → ω with injective and ω -almost-disjointfibers. Now, given k < ω and an increasing sequence h ξ n | n < ω + k i of countableordinals, we do the following. For each i < k , denote β i := ξ ω + i . • As p has ω -coherent fibers, a := S i
For every partition p : [ ω ] → ω and every uncountable X ⊆ ω such that p ↾ [ X ] does not witness U( ω , ω , ω, ω ) , for every coloring c : [ ω ] → ω with finite-to-one fibers, there exists j < ω such that { c ( α, β ) | ( α, β ) ∈ [ X ] , p ( α, β ) = j } is infinite . Proof. Suppose p and X are as above. Fix n, k < ω and an uncountable pairwisedisjoint family A ⊆ [ X ] k , such that for every uncountable B ⊆ A there is a pair( a, b ) ∈ [ B ] such that p [ a × b ] ∩ n = ∅ . By the Dushhnik-Miller theorem, then,there exists a < -increasing sequence h a i | i < ω + 1 i of elements of A such that p [ a i × a i ′ ] ∩ n = ∅ for all i < i ′ < ω + 1. It follows that there exist I ∈ [ ω ] ω , β ∈ a ω ,and h α i | i ∈ I i ∈ Q i ∈ I a i such that i p ( α i , β ) is constant over I with somevalue j < n . Then, for every coloring c : [ ω ] → ω with finite-to-one fibers, the set { c ( α, β ) | ( α, β ) ∈ [ X ] , p ( α, β ) = j } is infinite. (cid:3) The next result is Clause (3) of Theorem B. Corollary 4.10. Assuming MA ℵ ( K ) , for every partition p : [ ω ] → ω , the fol-lowing are equivalent: (1) ω → p [ ω ] ω,<ω ; (2) There exists X ∈ [ ω ] ℵ such that p ↾ [ X ] witnesses U( ω , ω , ω, ω ) .Proof. (1) = ⇒ (2): Fix any coloring c : [ ω ] → ω with finite-to-one fibers.Assuming that ω → p [ ω ] ω,<ω holds, let us now fix X ∈ [ ω ] ℵ that witnesses theinstance ω → p [ ω ] ω,<ω for the coloring c . This means that { c ( α, β ) | ( α, β ) ∈ [ X ] & p ( α, β ) = j } is finite for every j < ω . So, by Theorem 4.9, p ↾ [ X ] mustwitness U( ω , ω , ω, ω ).(2) = ⇒ (1): Fix X ∈ [ ω ] ℵ such that p ↾ [ X ] witnesses U( ω , ω , ω, ω ).Then, by Theorem 4.7, for every coloring c : [ ω ] → ω , there is a decomposition X = U i<ω X i such that, for all i, j < ω , { c ( α, β ) | ( α, β ) ∈ [ X i ] & p ( α, β ) = j } is finite. Find i < ω such that X i is uncountable. Then X i witnesses the instance ω → p [ ω ] ω,<ω for the coloring c . (cid:3) The same proof yields: Corollary 4.11. Assuming MA ℵ ( K ) , for every partition p : [ ω ] → ω , the fol-lowing are equivalent: (1) There is a decomposition ω = U i<ω X i such that, for all i, j < ω , { c ( α, β ) | ( α, β ) ∈ [ X i ] & p ( α, β ) = j } is finite ;(2) There is a decomposition ω = U i<ω X i such that, for all i < ω , p ↾ [ X i ] witnesses U( ω , ω , ω, ω ) . (cid:3) For completeness, we mention that by [CKS20, Corollary 29] it is consistent with MA ℵ ( σ -linked) that Pr ( ℵ , ℵ , ℵ , ℵ ) p holds for any partition p : [ ω ] → ω .5. Strong colorings versus a Luzin set It follows from Theorem 3.1 that the existence of a Luzin set gives rise to variousstrong colorings. Earlier results connecting Luzin sets with strong colorings include[EH78, Theorem 5.3] and [Tod87, p. 291]. Likewise, Souslin trees give rise to strong DVANCES ON STRONG COLORINGS OVER PARTITIONS 21 colorings, e.g., [Jen72, Lemma 6.6], [Tod81, Theorem 9.1] and it is a folklore factthat a coherent κ -Souslin tree (see [BR17, p. 1961]) entails Pr ( κ, κ, κ, ω ).In this section, we prove that a Luzin set and a coherent Souslin tree are notsufficient to yield strong colorings over all partitions. That is, we shall prove The-orem C: Theorem 5.1. It is consistent that all of the following hold simultaneously: • There exists Luzin set; • There exists a coherent Souslin tree; • There exists a partition p : [ ω ] → ω such that z p ( c ) ≤ ℵ for every coloring c : [ ω ] → ω . The model of Theorem 5.1 will be the outcome of a finite support iteration ofposets Q ( p, c ) of the following form. Definition 5.2. Q ( p, c ) consists of all triples q = ( a q , f q , w q ) satisfying all of thefollowing:(1) a q ∈ [ ω ] < ℵ ;(2) f q : a q → ω is a function;(3) w q is a function from a finite subset of ω × ω to ω ;(4) for all ( α, β ) ∈ [ a q ] , if f q ( α ) = f q ( β ), then ( f q ( α ) , p ( α, β )) ∈ dom( w q ) and c ( α, β ) = w q ( f q ( α ) , p ( α, β )).For a generic G ⊆ Q ( p, c ), let X i,G = { α < ω | ∃ q ∈ G ( f q ( α ) = i ) } . The proofof Theorem 4.5 makes it clear that for every partition p : [ ω ] → ω with injectiveand ω -almost-disjoint fibers, Q ( p, c ) has Property K , and for all i, j < ω ,1l (cid:13) Q ( p,c ) “ |{ c ( α, β ) | ( α, β ) ∈ [ X i, ˙ G ] and p ( α, β ) = j }| ≤ . Definition 5.3. For all q ∈ Q ( p, c ), k < ω and z ∈ [ ω ] < ℵ , define q ∧ ( k, z ) to bethe triple ( a, f, w ) satisfying: • a := a q ∪ z ; • f : a → ω is a function extending f q and satisfying f ( α ) = k + otp( z ∩ α )for all α ∈ a \ a q ; • w q := w .Note that q ∧ ( k, z ) may not be in Q ( p, c ), but it will be, provided that k ⊇ Im( f q ). Corollary 5.4. For every β < ω , D β := { q ∈ Q ( p, c ) | β ∈ a q } is dense, so that (cid:13) Q ( p,c ) “ ] i<ω X i, ˙ G = ω ” . Proof. Given arbitrary q ∈ Q ( p, c ) and β < ω , for all sufficiently large k , q ∧ ( k, { β } )is a condition in D β , extending q . (cid:3) Corollary 5.5. (cid:13) Q ( p,c ) “ z p ( c ) ≤ ℵ ”. (cid:3) Definition 5.6. Let p : [ ω ] → ω be a partition. For any ordinal η , a finite-support iteration { Q ξ } ξ ∈ η will be called a p -iteration iff Q is the trivial forcing,and, for each ordinal ξ with ξ + 1 < η there is a Q ξ -name ˙ c ξ such that(1) 1l (cid:13) Q ξ “ ˙ c ξ : [ ω ] → ω is a coloring”,(2) Q ξ +1 = Q ξ ∗ Q ( p, ˙ c ξ ). Convention . If { Q ξ } ξ ∈ η is a p -iteration, with η > Q η .From now on, we fix a p -iteration { Q ξ } ξ ∈ η for some partition p : [ ω ] → ω withinjective and ω -almost-disjoint fibers, hence each of the iterands has Property K ,and so does the whole iteration. Definition 5.8. A structure M is said to be good for the p -iteration { Q ξ } ξ ∈ η iffthere is a large enough regular cardinal κ > η such that all of the following hold: • M is a countable elementary submodel of ( H κ , ∈ , ⊳ κ ), where ⊳ κ is a well-ordering of H κ ; • p, { Q ξ } ξ ∈ η and { ˙ c ξ | ξ + 1 < η } are in M . Definition 5.9. Define q ∈ Q ξ to be determined by recursion on ξ ∈ η : ◮ For ξ = 0, all the conditions are determined. ◮ For any ξ , a condition q ∈ Q ξ +1 is determined if:(1) q ↾ ξ is determined;(2) q ↾ ξ (cid:13) Q ξ “ q ( ξ ) = ( a q,ξ , f q,ξ , w q,ξ )” for an actual triple of finite sets;(3) for all ( α, β ) ∈ [ a q,ξ ] there is some n < ω such that q ↾ ξ (cid:13) Q ξ “ ˙ c ξ ( α, β ) = n ”. ◮ For any ξ ∈ acc( η ), q ∈ Q ξ is determined if q ↾ ζ is determined for all ζ < ξ . Lemma 5.10. The determined conditions are dense in Q η .Proof. Standard. (cid:3) Definition 5.11. For a determined condition q in the p -iteration, we say that k is sufficiently large for q iff k ⊇ Im( f q,ξ ) for all ξ in the support of q . Definition 5.12. For a condition q in the p -iteration, k < ω and z ∈ [ ω ] < ℵ ,define q ∧ ( k, z ) by letting q ∧ ( k, z )( ξ ) := q ( ξ ) ∧ ( k, z ) for each ξ in the support of q .Note that if q is determined and k is sufficiently large for q , then for each ξ inthe support of q , q ↾ ξ (cid:13) Q ξ “ q ∧ ( k, z ) ∈ Q ( p, ˙ c ξ )”. In effect, if k is sufficiently largefor q , then q ∧ ( k, z ) is a legitimate condition. Definition 5.13. For any structure M good for the p -iteration { Q ξ } ξ ∈ η , for all ξ ∈ η and a determined condition q ∈ Q ξ , we define q M , as follows. The definitionis by recursion on ξ ∈ η : ◮ For ξ = 0 there is nothing to do. ◮ For any ξ such that q M has been defined for all determined q in Q ξ , given adetermined condition q ∈ Q ξ +1 , we consider two cases: ◮◮ If ξ ∈ M , then let q M := ( q ↾ ξ ) M ∗ ( a q,ξ ∩ M , f q,ξ ∩ M , w q,ξ ); ◮◮ Otherwise, just let q M := ( q ↾ ξ ) M ∗ ( ∅ , ∅ , ∅ ). ◮ For any ξ ∈ acc( η ), since this is a finite-support iteration, there is nothingnew to define.If q is determined, then, for every coordinate ξ in the support of q , q M ( ξ ) isa triple consisting of finite sets lying in M . It is important to note that q M maynot, in general, be a condition in Q ξ , because the last clause of Definition 5.2 mayfail. Nevertheless, ( q M ) ∧ ( k, z ) is a well-defined object, since its definition does notdepend on the ˙ c ξ ’s. DVANCES ON STRONG COLORINGS OVER PARTITIONS 23 Notation 5.14. For any determined condition q ∈ Q ξ , we denote by A q the unionof a q,ξ over all ξ in the support of q .We now arrive at the main technical lemma of this section. Lemma 5.15. Suppose p : [ ω ] → ω satisfies the conclusion of Lemma 2.9, and M is a structure which is good for the p -iteration { Q ξ } ξ ∈ η .For all ζ ≤ sup( η ) and a determined condition r ∈ Q ζ , there is a finite set ¯ z ⊆ M ∩ ω such that: A: For every z ∈ [ M ∩ ω ] < ℵ covering ¯ z and every integer k that is sufficientlylarge for r , ( r M ) ∧ ( k, z ) is in M ∩ Q ζ and is determined; B: For every z ∈ [ M ∩ ω ] < ℵ covering ¯ z and every integer k that is sufficientlylarge for r , for the condition ¯ r := ( r M ) ∧ ( k, z ) and a condition q ∈ M ∩ Q ζ ,if the following three requirements hold: (1) M | = q ≤ ¯ r and q is determined; (2) the mapping ( α, β ) p ( α, β ) is injective over ( A q \ A ¯ r ) × ( A r \ A ¯ r ) ; (3) p ( α, β ) > p ( α ′ , β ′ ) for all ( α, β ) ∈ ( A q \ A ¯ r ) × ( A r \ A ¯ r ) and ( α ′ , β ′ ) ∈ [ A r ] ∪ [ A q ] ,then q r .Proof. Proceed by induction on ζ ≤ sup( η ) proving A and B simultaneously. Thecase ζ = 0 is immediate. The case ζ = 1 is simple as well, but it may be instructiveto consider it in detail. So c is a coloring in the ground model and all conditionsare determined. In effect, given r ∈ Q , r M is a condition, as well. It will be shownthat ¯ z = ∅ satisfies the conclusion.Let k be sufficiently large for r . We know that ( r M ) ∧ ( k, z ) ∈ M ∩ Q for any z ∈ [ M ∩ ω ] < ℵ . Hence A is immediate. To see that B holds, suppose that we aregiven z ∈ [ M ∩ ω ] < ℵ , we let ¯ r := ( r M ) ∧ ( k, z ), and we are also given a condition q ∈ M ∩ Q satisfying requirements (1)–(3) above.To see that q r , let a := a q, ∪ a r, , f := f q, ∪ f r, and w := w q, ∪ w r, . It isimmediate to see that f and w are functions, A r = a r, , A q = a q, and A q ∩ A r = A ¯ r .We need to show that there exists a function w ∗ extending w for which ( a, f, w ∗ ) isa legitimate condition. For this, suppose that we are given i, j < ω , ( α, β ) , ( α ′ , β ′ ) ∈ [ a ] , with f ( α ) = f ( β ) = i = f ( α ′ ) = f ( β ′ ) and p ( α, β ) = j = p ( α ′ , β ′ ). It must beshown that c ( α, β ) = c ( α ′ , β ′ ). There are two cases to consider: Case I: If ( α, β ) , ( α ′ , β ′ ) ∈ [ A q ] ∪ [ A r ] , then since w extends w q, and w r, , c ( α, β ) = w ( i, j ) = c ( α ′ , β ′ ). Case II: If ( α, β ) ∈ [ a ] \ ([ A q ] ∪ [ A r ] ), then since A q ∩ A r = A ¯ r and α < β , we infer that ( α, β ) ∈ ( A q \ A ¯ r ) × ( A r \ A ¯ r ). So, by Clause (3),( α ′ , β ′ ) ∈ [ a ] \ ([ A q ] ∪ [ A r ] ), as well. Then, likewise ( α ′ , β ′ ) ∈ ( A q \ A ¯ r ) × ( A r \ A ¯ r ). Altogether, by Clause (2), ( α, β ) = ( α ′ , β ′ ). In particular, c ( α, β ) = c ( α ′ , β ′ ).Next, assume that ζ ≤ sup( η ) and that A and B have been established for all ξ < ζ . If ζ is a limit, then the finite-support nature of the iteration also establishesboth A and B hold, so suppose that ζ = ξ + 1. The successor case in which ξ / ∈ M also follows directly from the induction hypothesis by the definition of q M ,so assume that ξ ∈ M .Let r ∈ Q ζ be determined. Let ¯ z ∈ [ M ∩ ω ] < ℵ be given by the inductionhypothesis with respect to r ↾ ξ . In particular, for every z ∈ [ M ∩ ω ] < ℵ covering¯ z , and k sufficiently large for r ↾ ξ , (( r ↾ ξ ) M ) ∧ ( k, z ) is in M ∩ Q ξ and is determined. To establish A , note that, since ξ ∈ M , it follows that for any z ∈ [ M ∩ ω ] < ℵ covering ¯ z , and k sufficiently large for r (in particular, sufficiently large for r ↾ ξ ), s k,z := (( r ↾ ξ ) M ) ∧ ( k, z ) ∗ ( a r,ξ ∩ M , f r,ξ ∩ M , w r,ξ )is in M . It must also be shown that s k,z belongs to Q ζ . For this, it suffices to showthat for all i, j < ω ,(( r ↾ ξ ) M ) ∧ ( k, z ) (cid:13) Q ξ “ ∀ ( α, β ) ∈ [ f − r,ξ [ { i } ] ∩ M ] p ( α, β ) = j → ( ˙ c ξ ( α, β ) = w r,ξ ( i, j ))” . (1)Now note that for each ( α, β ) ∈ [ a r,ξ ∩ M ] there is a countable, maximal an-tichain deciding ˙ c ξ ( α, β ) and belonging to M because ˙ c ξ belongs to M ; in otherwords, all possible decisions about the value of ˙ c ξ ( α, β ) can be forced without leav-ing M . So, if assertion (1) fails, then there must be some q ∗ ≤ (( r ↾ ξ ) M ) ∧ ( k, z ) in M and ( α, β ) in [ f − r,ξ [ { i } ] ∩ M ] such that p ( α, β ) = j = p ( α ′ , β ′ ), but q ∗ (cid:13) Q ξ “ ˙ c ξ ( α, β ) = w r,ξ ( i, j )” . Fix k sufficiently large for r . Then, under the assumption that for any z ∈ [ M ∩ ω ] < ℵ covering ¯ z , s k,z does not belong to Q ζ , it is possible to constructrecursively a sequence { ( z n , q n , i n , j n , ( α n , β n ) } n ∈ ω such that: • z = ¯ z ; • q n ≤ (( r ↾ ξ ) M ) ∧ ( k, z n ) and q n is determined; • ( α n , β n ) ∈ [ f − r,ξ [ { i n } ] ∩ M ] ; • p ( α n , β n ) = j n ; • q n (cid:13) Q ξ “ ˙ c ξ ( α n , β n ) = w r,ξ ( i, j )”; • z n +1 ) A q n .Observe that by making canonical choices (e.g., by consulting with ⊳ κ ), this con-struction can be carried out in M . Let ǫ := sup( S n ∈ ω A q n ) + 1. Define a function h : ǫ → ω via h ( α ) := max { k, p ( α ′ , β ′ ) | ( α ′ , β ′ ) ∈ [ A q n +1 ] and α ∈ A q n +1 \ A q n } , and note that h is in M .Let γ satisfy the conclusion of Lemma 2.9 for h . As h ∈ M , γ ∈ M , so, since b := A r \ M is an element of [ ω \ γ ] < ℵ , there exists ∆ ∈ [ ǫ ] < ℵ such that: • p ↾ (( ǫ \ ∆) × b ) is injective; • for all α ∈ ǫ \ ∆ and β ∈ b , h ( α ) < p ( α, β ).Fix a large enough n < ω such that A q n +1 \ A q n is disjoint from ∆. Denote¯ r := (( r ↾ ξ ) M ) ∧ ( k, z n +1 ). As z n +1 ⊇ A q n , ( A q n +1 \ A ¯ r ) ⊆ ( A q n +1 \ A q n ) ⊆ ( ǫ \ ∆),( A r ↾ ξ \ A ¯ r ) ⊆ b , and all of the following hold:(1) M | = q n +1 ≤ ¯ r and q is determined;(2) the mapping ( α, β ) p ( α, β ) is injective over ( A q n +1 \ A ¯ r ) × ( A r ↾ ξ \ A ¯ r );(3) p ( α, β ) > p ( α ′ , β ′ ) for all ( α, β ) ∈ ( A q n +1 \ A ¯ r ) × ( A r ↾ ξ \ A ¯ r ) and ( α ′ , β ′ ) ∈ [ A r ↾ ξ ] ∪ [ A q n +1 ] .Then applying the induction hypothesis for B yields that q n +1 ( r ↾ ξ ). Picka determined condition q ∗ in Q ξ simultaneously extending q n +1 and ( r ↾ ξ ). As q ∗ ≤ q n +1 , we infer that q ∗ (cid:13) Q ξ “ ˙ c ξ ( α n +1 , β n +1 ) = w r,ξ ( i n +1 , j n +1 )” . DVANCES ON STRONG COLORINGS OVER PARTITIONS 25 As ( α n , β n ) ∈ [ f − r,ξ [ { i n } ] ∩ M ] , p ( α n , β n ) = j n and q ∗ ≤ r ↾ ξ , we infer that q ∗ (cid:13) Q ξ “ ˙ c ξ ( α n +1 , β n +1 ) = w r,ξ ( i n +1 , j n +1 )” . This is a contradiction. So A does hold.Next, let us establish B . Recall that we have a determined condition r ∈ Q ζ and ¯ z ∈ [ M ∩ ω ] < ℵ satisfying that for every z ∈ [ M ∩ ω ] < ℵ covering ¯ z , and k sufficiently large for r ↾ ξ , (( r ↾ ξ ) M ) ∧ ( k, z ) is in M ∩ Q ξ and is determined. Wehave just established A , proving that we may fix a finite z ∗ with ¯ z ⊆ z ∗ ⊆ M ∩ ω ,satisfying that for every z ∈ [ M ∩ ω ] < ℵ covering z ∗ , and every integer k that issufficiently large for r , ( r M ) ∧ ( k, z ) is in M ∩ Q ζ and is determined.Now, fix arbitrary z ∈ [ M ∩ ω ] < ℵ covering z ∗ , an integer k that is sufficientlylarge for r , and a condition q ∈ M ∩ Q ζ . Set ¯ r := ( r M ) ∧ ( k, z ) and suppose that therequirements (1)–(3) of B for ¯ r and q hold. In particular, they hold for ¯ r ↾ ξ and q ↾ ξ . That is: • M | = q ↾ ξ ≤ ¯ r ↾ ξ and q ↾ ξ is determined; • the mapping ( α, β ) p ( α, β ) is injective over ( A q ↾ ξ \ A ¯ r ↾ ξ ) × ( A r ↾ ξ \ A ¯ r ↾ ξ ); • p ( α, β ) > p ( α ′ , β ′ ) for all ( α, β ) ∈ ( A q ↾ ξ \ A ¯ r ↾ ξ ) × ( A r ↾ ξ \ A ¯ r ↾ ξ ) and ( α ′ , β ′ ) ∈ [ A r ↾ ξ ] ∪ [ A q ↾ ξ ] .Now, as z ∗ ⊇ ¯ z , we get from B of the previous stage that ( q ↾ ξ ) ( r ↾ ξ ). Picka determined condition q ∗ in Q ξ simultaneously extending ( q ↾ ξ ) and ( r ↾ ξ ). Let a := a q,ξ ∪ a r,ξ , f := f q,ξ ∪ f r,ξ and w := w q,ξ ∪ w r,ξ . It is immediate to see that f and w are functions, A r ⊇ a r,ξ , A q ⊇ a q,ξ and A q ∩ A r = A ¯ r . To see that q r , itsuffices to prove that there exists w ∗ ⊇ w such that q ∗ ∗ ( a, f, w ∗ ) ∈ Q ζ .For this, suppose that we are given i, j < ω , ( α, β ) , ( α ′ , β ′ ) ∈ [ a ] , with f ( α ) = f ( β ) = i = f ( α ′ ) = f ( β ′ ) and p ( α, β ) = j = p ( α ′ , β ′ ). It must be shown that q ∗ (cid:13) Q ξ “ ˙ c ξ ( α, β ) = ˙ c ξ ( α ′ , β ′ )” . There are two cases to consider: Case I: If ( α, β ) , ( α ′ , β ′ ) ∈ [ a q,ξ ] ∪ [ a r,ξ ] , then since w extends w q,ξ and w r,ξ ,the conclusion follows from the fact that q ∗ extends q ↾ ξ and r ↾ ξ . Case II: If ( α, β ) ∈ [ a ] \ ([ a q,ξ ] ∪ [ a r,ξ ] ), then, as seen earlier, requirements(2) and (3) imply that ( α, β ) = ( α ′ , β ′ ).So, we are done. (cid:3) Lemma 5.16. Suppose: • p : [ ω ] → ω satisfies the conclusion of Lemma 2.9; • L = { l γ } γ ∈ ω is a Luzin subset of ω ; • { Q ξ } ξ ∈ η is a p -iteration with η > a limit ordinal.Then (cid:13) Q η “ L is Luzin” . Proof. Suppose not. Then it can be assumed that there is a Q η -name ˙ T such that • (cid:13) Q η “ ˙ T ⊆ <ω is a closed nowhere dense tree”, and • (cid:13) Q η “( ∃ ℵ γ ) l γ is a branch through ˙ T ”.It follows that there is an uncountable subset Γ ⊆ ω such that for each γ ∈ Γ thereis a determined condition r γ ∈ Q η such that r γ (cid:13) Q η “ l γ is a branch through ˙ T ”.By possibly shrinking Γ, we may assume the existence of k < ω which is sufficiently large for r γ for all γ ∈ Γ, and we may also assume that { A r γ | γ ∈ Γ } forms a∆-system with some root ρ .Let M be a structure good for the p -iteration { Q ξ } ξ ∈ η , with ρ, ˙ T , Q η ∈ M .For each γ ∈ Γ, let ¯ z γ be given by Lemma 5.15 with respect to r γ . Fix anuncountable Γ ′ ⊆ Γ and some ¯ z ∈ [ ω ∩ M ] <ω such that ¯ z γ = ¯ z for all γ ∈ Γ ′ . Bypossibly shrinking further, we may assume the existence of q such that ( r γ ) M = q for all γ ∈ Γ ′ . In particular, for every z ∈ [ M ∩ ω ] < ℵ covering ¯ z , q ∧ ( k, z ) ∈ M ∩ Q η is determined.Let { τ n } n ∈ ω enumerate 2 <ω . construct recursively a sequence { ( z n , q n , t n ) } n ∈ ω such that: • z = ¯ z ∪ ρ ; • q n ≤ q ∧ ( k, z n ) and q n is a determined condition lying in M ; • τ n ⊆ t n ∈ <ω with q n (cid:13) Q η “ t n / ∈ ˙ T ”; • z n +1 ) A q n .Let ǫ := sup( S n ∈ ω A q n ) + 1. Define a function h : ǫ → ω via h ( α ) := max { k, p ( α ′ , β ′ ) | ( α ′ , β ′ ) ∈ [ A q n +1 ] and α ∈ A q n +1 \ A q n } . Recalling that p was given by Lemma 2.9, we now fix γ ∗ < ω satisfying that forevery b ∈ [ ω \ γ ∗ ] < ℵ , there exists ∆ ∈ [ ǫ ] < ℵ such that: • p ↾ (( ǫ \ ∆) × b ) is injective; • for all α ∈ ǫ \ ∆ and β ∈ b , h ( α ) < p ( α, β ).Clearly, Γ ∗ := { γ ∈ Γ ′ | min( A r γ \ ρ ) > γ ∗ } is uncountable. For each n < ω ,consider the open set U n := { l ∈ ω | t n ⊆ l } . Set W := T ∞ j =0 S ∞ j = n U n +1 . Then W is a dense G δ set, so since { l γ } γ ∈ Γ ∗ is Luzin, it is possible to find γ ∈ Γ ∗ suchthat l γ ∈ W . Set b := A r γ \ ρ and then let ∆ ∈ [ ǫ ] < ℵ be the corresponding set, asabove.Fix a large enough j < ω such that A q n +1 \ A q n is disjoint from ∆ for all n ≥ j . As l γ ∈ W , we may now fix some n ≥ j such that l γ ∈ U n +1 . Denote¯ r := ( q M ) ∧ ( k, z n +1 ). Then ( A q n +1 \ A ¯ r ) ⊆ ( A q n +1 \ A q n ) ⊆ ( ǫ \ ∆), ( A r γ \ A ¯ r ) ⊆ b ,and all of the following hold:(1) M | = q n +1 ≤ ¯ r and q is determined;(2) the mapping ( α, β ) p ( α, β ) is injective over ( A q n +1 \ A ¯ r ) × ( A r γ \ A ¯ r );(3) p ( α, β ) > p ( α ′ , β ′ ) for all ( α, β ) ∈ ( A q n +1 \ A ¯ r ) × ( A r γ \ A ¯ r ) and ( α ′ , β ′ ) ∈ [ A r γ ] ∪ [ A q n +1 ] .Since z n +1 ⊇ ¯ z and ¯ z was given by Lemma 5.15, we may apply B and infer that q n +1 r γ . However, q n +1 (cid:13) Q η “ t n +1 / ∈ ˙ T ” and r γ (cid:13) Q η “ l γ is a branch through ˙ T ”,contradicting the fact that t n +1 ⊆ l γ . (cid:3) Proof of Theorem 5.1. Start with a ground model V of GCH in which there exists acoherent Souslin tree. Using CH , fix a Luzin set L and a partition p as in Lemma 2.9.Let Q ω be the corresponding p -iteration, using H ℵ as our bookkeeping device ofnames of colorings ˙ c ξ , as in [Kun80, VIII, § ccc of the iteration implies that for every coloring c : [ ω ] → ω in the extension, there is a tail of ξ ∈ ω such that c admits a Q ξ -name in H ℵ of V . So, in the extension, for every coloring c : [ ω ] → ω , z p ( c ) ≤ ω .Finally, by Lemma 5.16, the Luzin set L survives, as well. (cid:3) DVANCES ON STRONG COLORINGS OVER PARTITIONS 27 Pump-up theorems for strong colorings over partitions This section contains six theorems which assure the existence of a stronger strongcoloring from a weaker one — over a partition. In all theorems the definitions of thestronger coloring from the given one does not depend on the partition but ratherworks for all partitions for which the weaker one works.The first few theorems deal with the problem of pumping up a coloring attaining λ many colors into one attaining λ + many. The classical stretching argument of λ colors to λ + colors employs a surjection e β : λ → β + 1 which depends on the largermember in an unordered pair (cf. [Tod87, p. 277]). Defining c + ( α, β ) := e β ( c ( α, β ))stretches a strong coloring c : [ λ + ] → λ to a strong coloring c + : [ λ + ] → λ + via apigeonhole consideration for stabilizing the stretch: for a prescribed color δ < λ + many e β will map the same i < λ to δ , and the original coloring c will indeedproduce any possible i < λ .The above one-dimensional stretching is incompatible with a two-dimensionalpartition. What we do next, then, is instead of letting c + ( α, β ) := e β ( c ( α, β )), welet c + ( α, β ) := e γ ( i ), where both γ and i are computed from the triple ( α, β, c ( α, β )).To motivate the statement of the first theorem of this section, notice that if λ + p [ λ + ] λ + holds, then so does λ + p [ λ + ] λ and λ + [ λ + ] λ + . The upcomingtheorem shows that it is possible to combine such two consequences — a witnessfor λ many colors over a partition with a witness for λ + many colors but not overa partition — into a single strong coloring. Theorem 6.1. Suppose ν, µ ≤ λ are cardinals and: • p : [ λ + ] → µ is a partition; • ν = 1 or ν = λ . More generally, cf([ λ ] ν , ⊆ ) ≤ λ suffices.If Pr ( λ + , λ ⊛ λ + (cid:30) ν ⊛ λ + , λ, χ ) p and Pr ( λ + , λ ⊛ λ + , λ + , χ ) both hold, then so does Pr ( λ + , λ ⊛ λ + (cid:30) ν ⊛ λ + , λ + , χ ) p .Proof. Fix a coloring c : [ λ + ] → λ which witnesses Pr ( λ + , λ ⊛ λ + (cid:30) ν ⊛ λ + , λ, χ ) p anda coloring d : [ λ + ] → λ + which witnesses Pr ( λ + , λ ⊛ λ + , λ + , χ ). For every β < λ + fix a surjection e β : λ → β + 1. Fix a bijection π : λ ↔ λ × λ . Define a coloring c + : [ λ + ] → λ + , as follows: For all α < β < λ , let c + ( α, β ) := 0; for α < β < λ + with β ≥ λ denote ( i, j ) := π ( c ( α, β )) and let: c + ( α, β ) := e d ( j,β ) ( i ) . To verify that c + witnesses Pr ( λ + , λ ⊛ λ + (cid:30) ν ⊛ λ + , λ + , χ ) p fix a pairwise disjointsubfamilies A , B ⊆ [ λ + ] <χ with |A| = λ and |B| = λ + . Denote B γj := { b ∈ B | min( b ) ≥ λ & d [ { j } × b ] = { γ }} . Claim 6.1.1. There exists j < λ for which { γ < λ + | |B γj | = λ + } is cofinal in λ + .Proof. Suppose not. Then, for every j < λ , δ j := sup { γ < λ + | |B γj | = λ + } is < λ + . Let δ := (sup j<λ δ j ) + 1. Then, for every j < λ + , |B δj | < λ + . In effect, B ′ := { b ∈ B | min( b ) ≥ λ & ∀ j < λ ( d [ { j } × b ] = { δ } ) } has size λ + . Appealing to d with A ′ := [ λ ] and B ′ , there must exist a ∈ A ′ and b ∈ B ′ such that d [ a × b ] = { δ } .But a = { j } for some j < λ , contradicting the fact that b ∈ B ′ . (cid:3) Let j < λ + be given by the claim. By the choice of c , for every γ < λ + such that |B γj | = λ + there exists A γ ∈ [ A ] ν such that for every function τ : µ → λ , there are a ∈ A γ and b ∈ B γj with a < b such that c ( α, β ) = τ ( p ( α, β )) for all ( α, β ) ∈ a × b .As { γ < λ + | |B γj | = λ + } is cofinal in λ + and cf([ |A| ] ν , ⊆ ) < λ + , we may find some A ′ ∈ [ A ] ν for which Γ := { γ < λ + | |B γj | = λ + & A γ ⊆ A ′ } is cofinal in λ + . Weclaim that A ′ is as sought. Claim 6.1.2. Let τ : µ → λ + . There are a ∈ A ′ and b ∈ B with a < b such that c ( α, β ) = τ ( p ( α, β )) for all ( α, β ) ∈ a × b .Proof. As µ ≤ λ , we may fix a large enough γ ∈ Γ such that Im( τ ) ⊆ γ . Forevery ǫ < µ , fix i ǫ < λ such that e γ ( i ǫ ) = τ ( ǫ ). Define a function τ ′ : µ → λ via τ ′ ( ǫ ) := π − ( i ǫ , j ). Pick a ∈ A γ and b ∈ B γj with a < b such that c ( α, β ) = τ ′ ( p ( α, β )) for all ( α, β ) ∈ a × b . Clearly, a ∈ A ′ and b ∈ B . Set ǫ := p ( α, β ). Then c ( α, β ) = τ ′ ( ǫ ) = π − ( i ǫ , j ), so that c + ( α, β ) = e d ( j,β ) ( i ǫ ) = e γ ( i ǫ ) = τ ( ǫ ). (cid:3) This completes the proof. (cid:3) The next result is Theorem D. It follows from the next three theorems, whichtogether cover all infinite cardinals λ . In the special case that λ is a singular cardinaland no partition p is taken into account, the result was first obtained by Eisworthas a corollary to his transformation theorem of [Eis13]. However, at the moment itis unclear whether such transformations can overcome partitions, hence the proofbelow is different. Theorem 6.2. For every coloring c : [ λ + ] → λ there exists a correspondingcoloring c + : [ λ + ] → λ + such that for every partition p : [ λ + ] → λ and everycardinal χ ≤ cf( λ ) : (1) if c witnesses Pr ( λ + , λ + , λ, χ ) p then c + witnesses Pr ( λ + , λ + , λ + , χ ) p ; (2) if c witnesses Pr ( λ + , λ + ⊛ λ + , λ, χ ) p then c + witnesses Pr ( λ + , λ + ⊛ λ + ,λ + , χ ) p .Proof. For λ regular, appeal to Theorem 6.4 below. For λ singular of countablecofinality, appeal to Theorem 6.3 below. For λ singular of uncountable cofinality,appeal to Theorem 6.5 below. (cid:3) Theorem 6.3. For every coloring c : [ λ + ] → λ there exists a correspondingcoloring c + : [ λ + ] → λ + such that for every partition p : [ λ + ] → µ with µ ≤ λ and every cardinal χ such that λ <χ = λ : (1) if c witnesses Pr ( λ + , λ + , λ, χ ) p then c + witnesses Pr ( λ + , λ + , λ + , χ ) p ; (2) if c witnesses Pr ( λ + , λ + ⊛ λ + , λ, χ ) p then c + witnesses Pr ( λ + , λ + ⊛ λ + ,λ + , χ ) p ; (3) if c witnesses Pr ( λ + , λ ⊛ λ + (cid:30) ν ⊛ λ + , λ, χ ) p with cf([ λ ] ν , ⊆ ) ≤ λ , then c + wit-nesses Pr ( λ + , λ ⊛ λ + (cid:30) ν ⊛ λ + , λ + , χ ) p .Proof. By the Engelking-Karlowicz theorem, fix a sequence h h j | j < λ i of functionsfrom λ + to λ with the property that for every a ⊆ λ + with λ | a | = λ and a function h : a → λ , there exists j < λ with h ⊆ h j . Define a function d : λ × λ + → λ + via d ( j, β ) := e β ( h j ( β )) . For every B ⊆ P ( λ + ), denote B γj := { b ∈ B | min( b ) ≥ λ & d [ { j } × b ] = { γ }} .The following is clear. Claim 6.3.1. Assuming λ <χ = λ , for every B ⊆ [ λ + ] <χ of size λ + , there exists j < λ for which { γ < λ + | |B γj | = λ + } is cofinal in λ + . (cid:3) DVANCES ON STRONG COLORINGS OVER PARTITIONS 29 The rest of the proof is very similar to that of Theorem 6.1. We fix a bijection π : λ ↔ λ × λ and, for every β < λ + , we fix a surjection e β : λ → β + 1. Givena coloring c : [ λ + ] → λ , we define the corresponding coloring c + : [ λ + ] → λ + byletting c + ( α, β ) := 0 for all α < β < λ and, given α < β < λ + with β ≥ λ , wedenote ( i, j ) := π ( c ( α, β )) and let: c + ( α, β ) := e d ( j,β ) ( i ) . The verification of the three clauses of this theorem is now similar to the verifi-cation in the proof of Theorem 6.1. (cid:3) The next theorem and the one following it will employ walks on ordinals in orderto pick the γ in the template formula “ c + ( α, β ) := e γ ( i )”. Theorem 6.4. Suppose that χ ≤ cf( λ ) and that E λ + ≥ χ admits a non-reflecting sta-tionary set. For every coloring c : [ λ + ] → λ there exists a corresponding coloring c + : [ λ + ] → λ + which satisfies that for every partition p : [ λ + ] → µ with µ ≤ λ : (1) if c witnesses Pr ( λ + , λ + , λ, χ ) p then c + witnesses Pr ( λ + , λ + , λ + , χ ) p ; (2) if c witnesses Pr ( λ + , λ + ⊛ λ + , λ, χ ) p then c + witnesses Pr ( λ + , λ + ⊛ λ + ,λ + , χ ) p .Proof. Fix a bijection π : λ ↔ λ × λ . For every β < λ + , fix a surjection e β : λ → β + 1. Fix a non-reflecting stationary set Γ ⊆ E λ + ≥ χ and a surjection h : λ + → λ + with the property that H γ := { α ∈ Γ | h ( α ) = γ } is stationary for all γ < λ + . Fixa sequence ~Z = h Z γ | γ < λ + i of elements of [ λ ] cf( λ ) such that, for all γ < δ < λ + , | Z γ ∩ Z δ | < cf( λ ).Let ~C = h C α | α < λ + i be a sequence such that, for every α < λ + C α is a closedsubset of α with sup( C α ) = sup( α ) and acc( C α ) ∩ Γ = ∅ . We shall be conductingwalks on ordinals along ~C , as follows (see [Tod07] for a comprehensive treatment).For all α < β < λ + , define a function Tr( α, β ) : ω → β + 1, by recursion on n < ω , as follows:Tr( α, β )( n ) := β, n = 0min( C Tr( α,β )( n − \ α ) , n > α, β )( n − > αα, otherwiseThen, derive a function ρ : [ λ + ] → ω via ρ ( α, β ) := min { n < ω | Tr( α, β )( n ) = α } . Now, given a coloring c : [ λ + ] → λ , we define a corresponding coloring c + :[ λ + ] → λ + , as follows. For every pair ( α, β ) ∈ [ λ + ] , first let ( i, ζ ) := π ( c ( α, β ));then, if there exists n < ω such that ζ ∈ Z Tr( α,β )( n ) , let c + ( α, β ) := e h (Tr( α,β )( n )) ( i )for the least such n . Otherwise, let c + ( α, β ) := 0.Next, let p : [ λ + ] → µ be an arbitrary partition with µ ≤ λ ; to see that c + is assought, assume one of the following:(1) c witnesses Pr ( λ + , λ + , λ, χ ) p and we are given a pairwise disjoint subfamily A of [ λ + ] <χ of size λ + , and a prescribed function τ : µ → λ + ;(2) c witnesses Pr ( λ + , λ + ⊛ λ + , λ, χ ) p and we are given two pairwise disjointsubfamilies A , B of [ λ + ] <χ of size λ + , and a prescribed function τ : µ → λ + . The proofs from either of the assumptions above are very similar. We will presentthem simultaneously, indicating by “Case (1)” and “Case (2)” the different parts.In case (2), for every α < λ + , pick a α ∈ A and b α ∈ B with min( x α ) > α , where x α := a α ∪ b α . In case (1), for every α < λ + , pick a α ∈ A with min( x α ) > α ,where x α := a α . Let D be some club in λ + such that, for every δ ∈ D and α < δ ,sup( x α ) < δ . Set γ := sup(Im( τ )). As µ ≤ λ , γ < λ + , so that ∆ := { δ ∈ D ∩ Γ | h ( δ ) = γ } is stationary. Next, define two functions f : ∆ → λ + and g : ∆ → λ via: • f ( δ ) := sup { sup( C Tr( δ,β )( i ) ∩ δ ) | β ∈ x δ , i < ρ ( δ, β ) } and • g ( δ ) := min( Z δ \ S { Z Tr( δ,β )( i ) | β ∈ x δ , i < ρ ( δ, β ) } .For every β ∈ x δ and i < ρ ( δ, β ), acc( C Tr( δ,β )( i ) ∩ Γ) = ∅ , so sup( C Tr( δ,β )( i ) ∩ δ ) < δ . It thus follows from | x δ | < · χ ≤ cf( δ ) that f ( δ ) < δ . Also, since | x δ | < cf( λ ), g ( δ ) is well-defined. Fix ( ξ ′ , ζ ′ ) ∈ λ + × λ for which ∆ ′ := { δ ∈ ∆ | f ( δ ) = ξ ′ & g ( δ ) = ζ ′ } is stationary.As the prescribed τ is a function from µ to γ + 1, we may fix, for every ǫ < µ , an i ǫ < λ such that e γ ( i ǫ ) = τ ( ǫ ). Define a function τ ′ : µ → λ via τ ′ ( ǫ ) := π − ( i ǫ , ζ ′ ).As ∆ ′ ⊆ D and | ∆ ′ | = λ + , we infer that A ′ := { a δ | δ ∈ ∆ ′ } is a subfamily of A ofsize λ + . Likewise, in Case (2), we also have that B ′ := { b δ | δ ∈ ∆ ′ } is a subfamilyof B of size λ + . So, in Case (1) (resp. Case (2)), we may fix a, b ∈ A ′ (resp. a ∈ A ′ and b ∈ B ′ ) with a < b such that c ( α, β ) = τ ′ ( p ( α, β )) for all α ∈ a and β ∈ b , Claim 6.4.1. Let ( α, β ) ∈ a × b . Then c + ( α, β ) = τ ( p ( α, β )) .Proof. Denote ǫ := p ( α, β ). By the definition of τ ′ , c ( α, β ) = τ ′ ( ǫ ) = π − ( i ǫ , ζ ′ ).By the choice of b , let us fix δ ∈ ∆ ′ such that b ⊆ x δ . As β ∈ x δ , ξ ′ = f ( δ ) <δ < β . Likewise, since α ∈ a ∈ A ′ , ξ ′ < α . Altogether,max { sup( C Tr( δ,β )( i ) ∩ δ ) | i < ρ ( δ, β ) } ≤ f ( δ ) = ξ ′ < α < δ < β. Now, by a standard fact from the theory of walks on ordinals (see [Rin14b, Claim 3.1.2]),Tr( α, β )( i ) = Tr( δ, β )( i ) for all i < ρ ( δ, β ), and Tr( α, β )( ρ ( δ, β )) = δ . Recall-ing that g ( δ ) = ζ ′ , this means that n := ρ ( δ, β ) is the least integer for which ζ ′ ∈ Z Tr( α,β )( n ) . Therefore, by the definition of c + , c + ( α, β ) = e h (Tr( α,β )( n )) ( i ǫ ) = e h ( δ ) ( i ǫ ) = e γ ( i ǫ ) = τ ( ǫ ) , as sought. (cid:3) This completes the proof. (cid:3) Unlike the preceding theorem, the proof of the next does not employ a function h : λ + → λ + , since it is still open whether for every singular cardinal λ there is a ~C -sequence that gives rise to a decomposition of λ + into λ + many walk-wise-largesets. In the template formula “ c + ( α, β ) := e γ ( i )”, instead of letting γ := h ( δ ) forsome well-chosen δ in the walk from β down to α , we shall let γ be the ξ th elementof C δ , for well-chosen δ in the walk and ξ < λ . Theorem 6.5. Suppose that λ is a singular cardinal of uncountable cofinality and χ ≤ cf( λ ) . For every coloring c : [ λ + ] → λ , there exists a corresponding coloring c + : [ λ + ] → λ + satisfying that for every partition p : [ λ + ] → µ with µ ≤ λ : (1) if c witnesses Pr ( λ + , λ + , λ, χ ) p then c + witnesses Pr ( λ + , λ + , λ + , χ ) p ; (2) if c witnesses Pr ( λ + , λ + ⊛ λ + , λ, χ ) p then c + witnesses Pr ( λ + , λ + ⊛ λ + ,λ + , χ ) p . DVANCES ON STRONG COLORINGS OVER PARTITIONS 31 Proof. By the proof of Case 1 of Theorem 4.21 from [LR18], we may fix a C -sequence ~C = h C α | α < λ + i such that otp( C α ) < λ for all α < λ + and such thatthe functions Tr and ρ derived from walking along ~C (as defined in the proof ofTheorem 6.4) satisfy the following. Claim 6.5.1. Let X be a pairwise disjoint subfamily of [ λ + ] < cf( λ ) of size λ + . Thenthere exists a stationary set ∆ ⊆ λ + , a sequence h x γ | γ ∈ ∆ i , and an ordinal ε < λ + , such that, for every γ ∈ ∆ : • x γ ∈ X with min( x γ ) > γ > ε ; • for all α ∈ ( ε, γ ) and β ∈ x γ , γ ∈ Im(Tr( α, β )) ; • cf( γ ) > sup { otp( C Tr( γ,β )( n ) ) | β ∈ x γ , n < ρ ( γ, β ) } .Proof. It suffices to prove that for every club D ⊆ λ + there are γ ∈ D , x γ ∈ X and an ordinal ε < γ such that the three bullets above hold. Now, given anarbitrary club D ⊆ λ + , Claim 4.21.2 from [LR18] provides γ ∈ D , x γ ∈ X andan ordinal ε < γ such that the first two bullets hold. The proof of that claimmakes it clear that cf( γ ) > | C | where C := S { C Tr( γ,β )( n ) | β ∈ x γ , n ≤ ρ ( γ, β ) } ,and goes through even if we require cf( γ ) > | C | + . In particular, this will givecf( γ ) > sup { otp( C Tr( γ,β )( n ) ) | β ∈ x γ , n < ρ ( γ, β ) } . (cid:3) Fix a bijection π : λ ↔ λ × λ . For every β < λ + , fix a surjection e β : λ → β + 1.Now, given a coloring c : [ λ + ] → λ , we define a corresponding coloring c + : [ λ + ] → λ + as follows. For every pair ( α, β ) ∈ [ λ + ] , first let ( i, ξ ) := π ( c ( α, β )), and then,if there exists n < ω such that otp( C Tr( α,β )( n ) ) > ξ , let c + ( α, β ) := e C Tr( α,β )( n ) ( ξ ) ( i )for the least such n . Otherwise, just let c + ( α, β ) := 0.Next, let p : [ λ + ] → µ be an arbitrary partition with µ ≤ λ ; to see that c + is assought, there are two cases to consider:(1) Assume c witnesses Pr ( λ + , λ + , λ, χ ) p , and we are given a pairwise disjointsubfamily A of [ λ + ] <χ of size λ + , and a prescribed function τ : µ → λ + .In this case, appeal to Claim 6.5.1 with X := A , and obtain a stationaryset ∆ ⊆ λ + , a sequence h x γ | γ ∈ ∆ i and an ordinal ε < λ + .(2) Assume c witnesses Pr ( λ + , λ + ⊛ λ + , λ, χ ) p , and we are given two pairwisedisjoint subfamilies A , B of [ λ + ] <χ of size λ + , and a prescribed function τ : µ → λ + .In this case, appeal to Claim 6.5.1 with some pairwise disjoint subfamily X of [ λ + ] < cf( λ ) of size λ + such that, for every x ∈ X , there are a ∈ A and b ∈ B such that x = a ⊎ b . In return, we obtain a stationary set ∆ ⊆ λ + , asequence h x γ | γ ∈ ∆ i and an ordinal ε < λ + . Then, for every γ ∈ ∆, fix a γ ∈ A and b γ ∈ B such that x γ = a γ ⊎ b γ .Next, let D be some club in λ + such that, for every δ ∈ D and α ∈ ∆ ∩ δ ,sup( x α ) < δ . By shrinking D , we may also assume that min( D ) > sup(Im( τ )). Forevery δ ∈ D ∩ ∆, let ξ δ denote the least ordinal ξ < λ withsup { otp( C Tr( δ,β )( n ) ) | β ∈ x δ , n < ρ ( δ, β ) } < ξ < otp( C δ )such that C δ ( ξ ) > sup(Im( τ )). Fix ξ < λ and γ < λ + for which ∆ ′ := { δ ∈ ∆ ∩ D | ξ δ = ξ & C δ ( ξ ) = γ } . As the prescribed τ is a function from µ to γ , for every ǫ < µ , we may fix i ǫ < λ such that e γ ( i ǫ ) = τ ( ǫ ). Define a function τ ′ : µ → λ via τ ′ ( ǫ ) := π − ( i ǫ , ξ ). As ∆ ′ ⊆ D and | ∆ ′ | = λ + , we infer that A ′ := { a δ | δ ∈ ∆ ′ } is a subfamily of A of size λ + . Likewise, in Case (2), we also have that B ′ := { b δ | δ ∈ ∆ ′ } is a subfamily of B of size λ + . So, in Case (1) (resp. Case (2)), we may fix a, b ∈ A ′ (resp. a ∈ A ′ and b ∈ B ′ ) with a < b such that c ( α, β ) = τ ′ ( p ( α, β )) forall α ∈ a and β ∈ b , Claim 6.5.2. Let ( α, β ) ∈ a × b . Then c + ( α, β ) = τ ( p ( α, β )) .Proof. Denote ǫ := p ( α, β ). By the definition of τ ′ , π ( c ( α, β )) = π ( τ ′ ( ǫ )) = ( i ǫ , ξ ).By the choice of b , let us fix δ ∈ ∆ ′ such that b ⊆ x δ . As β ∈ x δ and α ∈ a ∈ A ′ , ε < α < δ < β, so that δ ∈ Im(Tr( α, β )). Now, by the same standard fact used in the proof ofClaim 6.4.1, Tr( α, β )( i ) = Tr( δ, β )( i ) for all i < ρ ( δ, β ), and Tr( α, β )( ρ ( δ, β )) = δ .Recalling the choice of ξ , this means that n := ρ ( δ, β ) is the least integer to satisfyotp( C Tr( α,β )( n ) ) > ξ . So, by the definition of c + , we infer that c + ( α, β ) = e C Tr( α,β )( n ) ( ξ ) ( i ) = e C δ ( ξ ) ( i ǫ ) = e γ ( i ǫ ) = τ ( ǫ ) , as sought. (cid:3) This completes the proof. (cid:3) The next result is Theorem E, and it assumes the conventions established inSubsection 1.2. Its proof follows Shelah’s proof of the pump-up from Pr to Pr [She94, Lemma 4.5] and adds to it considerations to handle the partition. Thisanswers [CKS20, Question 46] in the affirmative. Theorem 6.6. Assume λ <χ < κ ≤ λ and λ <χ ≤ θ <χ = θ . For every coloring c : [ κ ] → θ , there exists a corresponding coloring c : [ κ ] → θ satisfying that forevery partition p : [ κ ] → µ : (1) if c witnesses Pr ( κ, κ, θ, χ ) p , then c witnesses Pr ( κ, κ, θ, χ ) p ; (2) if c witnesses Pr ( κ, ν ⊛ κ (cid:30) ⊛ κ , θ, χ ) p , then c witnesses Pr ( κ, ν ⊛ κ (cid:30) ⊛ κ , θ, χ ) p .Proof. As κ ≤ λ , let h X α | α < κ i be an injective sequence of subsets of λ . Claim 6.6.1. For every σ < χ and a ∈ [ κ ] σ , there are y ∈ [ λ ] <χ and an injection f : σ → P ( y ) , such that, for all α ∈ a , X α ∩ y = f (otp( α ∩ a )) .Proof. For all α < β < κ , let δ α,β := min( X α △ X β ). Now, let y := { δ ( α, β ) | α, β ∈ a, α = β } and then define a function f : σ → P ( y ) via f ( i ) := X a ( i ) ∩ y .Evidently, y and f are as required. (cid:3) Consider the following set: W := { ( y , y , Z , g ) | y , y ∈ [ λ ] <χ , Z ∈ [ y ∪ y ] <χ , g ∈ Z×Z θ } . It is clear that | W | = θ , so let us fix an enumeration h ( y j , y j , Z j , g j ) | j < θ i of W .For all j < θ , define a function h j : [ κ ] → θ via: h j ( α, β ) := ( g j ( X α ∩ y j , X β ∩ y j ) if X α ∩ y j ∈ Z j and X β ∩ y j ∈ Z j ;0 otherwise . Finally, given c : [ κ ] → θ , define another coloring, c : [ κ ] → θ , via c ( α, β ) := h c ( α,β ) ( α, β ) . (1) Suppose that c witnesses Pr ( κ, κ, θ, χ ) p . To see that c witnesses Pr ( κ, κ, θ, χ ) p , DVANCES ON STRONG COLORINGS OVER PARTITIONS 33 fix an arbitrary σ < χ , a κ -sized pairwise disjoint family A ⊆ [ κ ] σ and a matrix( τ ξ,ζ ) ξ,ζ<σ of functions from µ to θ . By Claim 6.6.1 and a pigeonhole argument,we may assume the existence of a set y ∈ [ λ ] <χ and an injection f : σ → P ( y ) suchthat, for all a ∈ A and α ∈ a , X α ∩ y = f (otp( α ∩ a )).Denote Z := Im( f ). For every ǫ < µ , define a function g ǫ : Z × Z → θ via g ǫ ( f ( ξ ) , f ( ζ )) := τ ξ,ζ ( ǫ ) . Then pick j ǫ < θ such that ( y j ǫ , y j ǫ , Z j ǫ , g j ǫ ) = ( y, y, Z , g ǫ ). Finally, define τ ∗ : µ → θ via τ ∗ ( ǫ ) := j ǫ and then pick ( a, b ) ∈ [ A ] such that c ( α, β ) = τ ∗ ( p ( α, β ))for all α ∈ a and β ∈ b . Claim 6.6.2. Let ξ, ζ < σ . Then c ( a ( ξ ) , b ( ζ )) = τ ξ,ζ ( p ( a ( ξ ) , b ( ζ )) .Proof. Write ǫ := p ( a ( ξ ) , b ( ζ )). Then c ( a ( ξ ) , b ( ζ )) = τ ∗ ( p ( a ( ξ ) , b ( ζ ))) = τ ∗ ( ǫ ) = j ǫ . Altogether c ( a ( ξ ) , b ( ζ )) = h j ǫ ( a ( ξ ) , b ( ζ )) = g j ǫ ( X a ( ξ ) ∩ y j ǫ , X b ( ζ ) ∩ y j ǫ ) = g ǫ ( f ( ξ ) , f ( ζ )) = τ ξ,ζ ( ǫ ) , as sought. (cid:3) (2) Suppose that c witnesses Pr ( κ, ν ⊛ κ (cid:30) ⊛ κ , θ, χ ) p . To see that c witnessesPr ( κ, ν ⊛ κ (cid:30) ⊛ κ , θ, χ ) p , fix an arbitrary σ < χ , a ν -sized pairwise disjoint family A ⊆ [ κ ] σ and a κ -sized pairwise disjoint family B ⊆ [ κ ] σ . By Claim 6.6.1, we mayassume the existence of a set y ∈ [ λ ] <χ and an injection f : σ → P ( y ), suchthat, for all b ∈ B and β ∈ B , X β ∩ y = f (otp( β ∩ b )). Now, by the hypothesison c , fix a ∈ A such that, for every τ : µ → θ , there exist b ∈ B with a < b suchthat, for all α ∈ a and β ∈ b , c ( α, β ) = τ ( p ( α, β )).By Claim 6.6.1, fix y ∈ [ λ ] <χ and an injection f : σ → P ( y ) such that, for all α ∈ a , X α ∩ y = f (otp( α ∩ a )). Denote Z := Im( f ) ∪ Im( f ).Now, given a matrix ( τ ξ,ζ ) ξ,ζ<σ of functions from µ to θ , for every ǫ < µ , pickany function g ǫ : Z × Z → θ satisfying that for all ξ, ζ < σ : g ǫ ( f ( ξ ) , f ( ζ )) = τ ξ,ζ ( ǫ ) . Then pick j ǫ < θ such that ( y j ǫ , y j ǫ , Z j ǫ , g j ǫ ) = ( y , y , Z , g ǫ ). 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