Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC
aa r X i v : . [ m a t h . L O ] S e p MAXIMAL INDEPENDENT SETS, VARIANTS OF CHAIN/ANTICHAINPRINCIPLE AND COFINAL SUBSETS WITHOUT AC
AMITAYU BANERJEE
Abstract.
In set theory without the Axiom of Choice (AC), we observe new relations of thefollowing statements with weak choice principles. • P lf,c (Every locally finite connected graph has a maximal independent set). • P lc,c (Every locally countable connected graph has a maximal independent set). • CAC ℵ α (If in a partially ordered set all antichains are finite and all chains have size ℵ α ,then the set has size ℵ α ) if ℵ α is regular. • CWF (Every partially ordered set has a cofinal well-founded subset). • If G = ( V G , E G ) is a connected locally finite chordal graph, then there is an ordering < of V G such that { w < v : { w, v } ∈ E G } is a clique for each v ∈ V G . introduction As usual, ZF denotes the Zermelo-Fraenkel set theory without the Axiom of Choice (AC), andZFA is ZF with the axiom of extensionality weakened to allow the existence of atoms. In thisnote, we observe new relations of some combinatorial statements with weak choice principles.1.1.
Maximal independent sets.
Friedman [[Fri11],
Theorem 6.3.2, Theorem 2.4 ] provedthat AC is equivalent to the statement ‘Every graph has a maximal independent set’ (abbreviatedhere as P ) in ZF. Spanring [Spa14] gave a different argument to prove the result. Consider thefollowing weaker formulations of P . • Fix n ∈ ω \{ , } . We denote by P K n , the class of those graphs whose only componentsare K n (complete graph on n vertices). We denote by P n the statement ‘Every graphfrom the class P K n has a maximal independent set’ . • We denote by P lf,c the statement ‘Every locally finite connected graph has a maximalindependent set’ . • We denote by P lc,c the statement ‘Every locally countable connected graph has a maximalindependent set’ .In this note, we observe the following.(1) AC n (Every family of n element sets has a choice function) is equivalent to P n for every n ∈ ω \{ , } in ZF (c.f. [ § , Observation 3.1 ]).(2) AC ωfin (Every denumerable family of non-empty finite sets has a choice function) isequivalent to P lf,c in ZF (c.f. [ § , Observation 3.2 ]).(3)
U T ( ℵ , ℵ , ℵ ) (The union of any countable family of countable sets is countable) implies P lc,c , and P lc,c implies AC ℵ ℵ (Every denumerable family of denumerable sets has a choicefunction) in ZF (c.f. [ § , Observation 3.3 ]).1.2.
A variant of Chain/Antichain principle.
A famous application of the infinite Ramsey’stheorem is the
Chain/Antichain principle (abbreviated here as “CAC”), which states that ‘Anyinfinite partially ordered set contains either an infinite chain or an infinite antichain’ . Tachtsis
Key words and phrases.
Maximal Independent sets, Variants of chain/antichain principle, Cofinal well-foundedsubsets of partially ordered sets, Fraenkel-Mostowski (FM) permutation models of ZFA+ ¬ AC . [Tac16] investigated the possible placement of CAC in the hierarchy of weak choice principles.Komj´ath–Totik [KT06] proved the following generalized versions of CAC, applying Zorn’s lemma. • If in a partially ordered set all antichains are finite and all chains are countable, thenthe set is countable (c.f. [[KT06],
Chapter 11 , Problem 8 ]). • If in a partially ordered set all chains are finite and all antichains are countable, thenthe set is countable (c.f. [[KT06],
Chapter 11 , Problem 7 ]).For each regular ℵ α , we denote by CAC ℵ α the statement ‘if in a partially ordered set all an-tichains are finite and all chains have size ℵ α , then the set has size ℵ α ’ and we denote by CAC ℵ α the statement ‘if in a partially ordered set all chains are finite and all antichains have size ℵ α ,then the set has size ℵ α ’ . In [BG20], we observed that for any regular ℵ α and any 2 ≤ n < ω , CAC ℵ α does not imply AC − n (Every infinite family of n -element sets has a partial choice func-tion) in ZFA. In [BG20], we also observed that CAC ℵ α does not imply ‘there are no amorphoussets’ in ZFA. In this note, we observe the following.(1) For any regular ℵ α , and any 2 ≤ n < ω , CAC ℵ α does not imply AC − n in ZFA (c.f.[ § , Theorem 4.3 ]). In particular, for any regular ℵ α , CAC ℵ α holds in the modelconstructed in the proof of [[HT20], Theorem 8 ].(2) For any regular ℵ α , CAC ℵ α does not imply ‘there are no amorphous sets’ in ZFA (c.f.[ § , Theorem 4.4 ]). In particular, for any regular ℵ α , CAC ℵ α holds in the basicFraenkel model.(3) CAC ℵ implies P AC ℵ fin in ZF if we denote by P AC ℵ fin the statement ‘Every infinite ℵ -sized family A of non-empty finite sets has a ℵ -sized subfamily B with a choice function’ (c.f. [ § , Theorem 4.6 ]).(4) DC (Dependent choice) does not imply
CAC ℵ in ZFA (c.f. [ § , Theorem 4.7 ]).1.3.
Cofinal well-founded subsets and improving the choice strength of a result.
Tachtsis [[Tac18],
Theorem 10(ii) ] proved that CWF (Every partially ordered set has a cofinalwell-founded subset) holds in the basic Fraenkel model. In [[THS16],
Theorem 3.26 ], Tachtsis,Howard, and Saveliev proved that CS (Every partially ordered set without a maximal elementhas two disjoint cofinal subsets) holds in the basic Fraenkel model. Halbeisen–Tachtsis [[HT20],
Theorem 10(ii) ] proved that LOC − (Every infinite linearly orderable family of 2-element setshas a partial choice function) does not imply LOKW − (Every infinite linearly orderable family A of 4-element sets has a partial Kinna–Wegner selection function) in ZFA. We construct amodel of ZFA and observe the following.(1) (LOC − + CS + CWF) does not imply LOC − n in ZFA if n ∈ ω such that n = 3 or n > § , Theorem 5.2 ]).(2) (LOC − + CS + CWF) does not imply CAC ℵ in ZFA (c.f. [ § , Corollary 5.3 ]).Fix n ∈ ω \{ , } , and k ∈ ω \{ , , } . The authors of [CHHKR08] proved that AC n holds in N ∗ ( k ) (generalised version of Howard’s model N ∗ (3) from [HR98]) if k has no divisors less thanor equal to n (c.f. [[CHHKR08], Theorem 4.8 ]). We observe that it is possible to improve thechoice strength of the result if k is a prime applying the methods of [HT13]. In particular, weobserve the following.(1) Fix any prime p ∈ ω \{ , , } , and any n ∈ ω \{ , } . If p is not a divisor of n , then AC n holds in N ∗ ( p ). Moreover, CWF holds in N ∗ ( p ) (c.f. [ § , Theorem 5.4 ]).We also remark that CWF holds in the Second Fraenkel’s model (labeled as Model N in [HR98]),and N ( p ) (the model from [[HT13], § ]) for any prime p ∈ ω \{ , , } (c.f. [ § , Remark5.5 , Remark 5.6 ]).1.4.
Locally finite connected chordal graphs.
Fulkerson–Gross [FG65] proved that a finitegraph G = ( V G , E G ) is chordal if and only if there is an ordering < of V G such that { w < v : AXIMAL INDEPENDENT SETS, VARIANTS OF CAC, AND CWF 3 { w, v } ∈ E G } is a clique for each v ∈ V G (c.f. [[Kom15], Lemma 1 ]). We apply the result toobserve the following.(1) AC ωfin implies the statement ‘If G = ( V G , E G ) is a connected locally finite chordal graph,then there is an ordering < of V G such that { w < v : { w, v } ∈ E G } is a clique for each v ∈ V G ’ in ZF (c.f. [ § , Observation 3.5 ]).We also list some other graph-theoretical statements restricted to locally finite connected graphs,which follows from AC ωfin in ZF (c.f. [ § , Remark 3.6 ]).2.
Notations, definitions, and known results
Definition 2.1. (Graph-theoretical definitions, and notations) . A graph G = ( V G , E G )is locally finite if every vertex of G has finite degree. We say that a graph is locally countable ifevery vertex has denumerable set of neighbours. Given a non-negative integer n , a path of length n in the graph G = ( V G , E G ) is a one-to-one finite sequence { x i } ≤ i ≤ n of vertices such that foreach i < n , { x i , x i +1 } ∈ E G ; such a path joins x to x n . The graph G is connected if any twovertices are joined by a path of finite length. An independent set is a set of vertices in a graph,no two of which are connected by an edge. A set W G ⊆ V G is called a maximal independent set in G = ( V G , E G ) if and only if it is independent and there is no independent set W ′ G such that W G ⊆ W ′ G (c.f.[Spa14]). A clique is a set of vertices in a graph, such that any two of them arejoined by an edge. We denote by K n , the complete graph on n vertices. Definition 2.2. (Chain, antichain, cofinal well-founded subsets) . Let P be a set. Abinary relation ≤ on P is called a partial order on P if ≤ is reflexive, antisymmetric, andtransitive. The ordered pair ( P, ≤ ) is called a partially ordered set or poset . A subset D ⊆ P is called a chain if ( D, ≤ ↾ D ) is linearly ordered. A subset A ⊆ P is called an antichain if notwo elements of A are comparable under ≤ . A subset C ⊆ P is called cofinal in P if for every x ∈ P there is an element c ∈ C such that x ≤ c . An element p ∈ P is minimal if for all q ∈ P ,( q ≤ p ) implies ( q = p ). A subset W ⊆ P is well-founded if every non-empty subset V of W hasa ≤ -minimal element. Definition 2.3. (Amorphous sets).
An innite set X is called amorphous if X cannot bewritten as a disjoint union of two innite subsets. Definition 2.4. (A list of forms). (1) The
Axiom of Choice , AC (Form 1 in [HR98] ) : Every family of nonempty sets hasa choice function.(2) AC ωfin (Form 10 in [HR98] ) : Every denumerable family of non-empty finite sets has achoice function. We recall two equivalent formulations of AC ωfin . • U T ( ℵ , f in, ℵ ) (Form 10 A in [HR98] ) : The union of denumerably many pairwisedisjoint finite sets is denumerable. • P AC ωfin (Form 10 E in [HR98] ) : Every denumerable family of finite sets has aninfinite subfamily with a choice function.(3) AC ℵ ℵ (Form 32 A in [HR98] ) : Every denumerable set of denumerable sets has a choicefunction. We recall the following equivalent formulation of AC ℵ ℵ . • P AC ℵ ℵ (Form 32 B in [HR98] ) : Every denumerable set of denumerable sets hasan infinite subset with a choice function.(4) AC (Form 88 in [HR98] ) : Every family of pairs has a choice function.(5) AC n for each n ∈ ω, n ≥ (Form 61 in [HR98] ) : Every family of n element setshas a choice function. We denote by AC − n the statement ‘Every infinite family of n -element sets has a partial choice function’ ( Form 342(n) in [HR98] , denoted by C − n in Definition 1 (2) of [HT20] ). (6) LOC − n for each n ∈ ω, n ≥ (see [HT20] ) : Every infinite linearly orderable familyof n -element sets has a partial choice function. We denote by LOKW − n the statement AMITAYU BANERJEE ‘Every infinite linearly orderable family A of n -element sets has a partial Kinna–Wegnerselection function’ (c.f. Definition 1 (2) of [HT20] ). (7) The Van Douwens Choice Principle , vDCP (see [HT13]): Every family X = { ( X i , ≤ i ) : i ∈ I } of linearly ordered sets isomorphic with ( Z , ≤ ) ( ≤ is the usual orderingon Z ) has a choice function.(8) The Axiom of Multiple Choice , MC (Form 67 in [HR98] ) : Every family A ofnon-empty sets has a multiple choice function, i.e., there is a function f with domain A such that for every A ∈ A , f ( A ) is a non-empty finite subset of A .(9) MC(n) where n ≥ is an integer (see [HT13] ) : For every family { X i : i ∈ I } ofnon-empty sets, there is a function F with domain I such that for all i ∈ I , we havethat F ( i ) is a finite subset of X i and gcd ( n, | F ( i ) | ) = 1.(10) LW (Form 90 in [HR98] ) : Every linearly-ordered set can be well-ordered.(11) AC W O (Form 40 in [HR98] ) : Every well-ordered set of non-empty sets has a choicefunction.(12) DC κ for an innite well-ordered cardinal κ (Form 87( κ ) in [HR98] ) : Let κ bean innite well-ordered cardinal (i.e., κ is an aleph). Let S be a non-empty set and let R be a binary relation such that for every α < κ and every α -sequence s = ( s ǫ ) ǫ<α ofelements of S there exists y ∈ S such that sRy . Then there is a function f : κ → S such that for every α < κ , ( f ↾ α ) Rf ( α ). We note that DC ℵ is a reformulation of DC(the principle of Dependent Choices (Form 43 in [HR98] ) ). We denote by DC <λ theassertion ( ∀ η < λ ) DC η .(13) UT(WO, WO, WO) (Form 231 in [HR98] ) : The union of a well-ordered collectionof well-orderable sets is well-orderable.(14) ( ∀ α ) U T ( ℵ α , ℵ α , ℵ α ) (Form 23 in [HR98] ) : For every ordinal α , if A and every memberof A has cardinality ℵ α , then | ∪ A | = ℵ α .(15) ℵ is regular (Form 34 in [HR98] ) .(16) Dilworths decomposition theorem for infinite posets of finite width,
DT (c.f. [Tac19] ): If P is an arbitrary poset, and k is a natural number such that P has no antichains of size k + 1 while at least one k -element subset of P is an antichain, then P can be partitionedinto k chains.(17) The Chain/Antichain Principle , CAC (Form 217 in [HR98] ) : Every infinite posethas an infinite chain or an infinite antichain.(18) There are no amorphous sets (Form 64 in [HR98] ) .(19) CS (see [THS16]): Every poset without a maximal element has two disjoint cofinalsubsets.(20) CWF (see [Tac18]): Every poset has a cofinal well-founded subset.(21)
A weaker form of Lo´s’s lemma, LT (Form 253 in [HR98] ) : If A = h A, R A i isa non-trivial relational L -structure over some language L , and U be an ultrafilter on anon-empty set I , then the ultrapower A I / U and A are elementarily equivalent.2.1. Group-theoretical facts.
A group G acts on a set X if for each g ∈ G there is a mapping x → gx of X into itself, such that 1 x = x for every x ∈ X and h ( gx ) = ( hg ) x for every g, h ∈ G .Alternatively, actions of a group G on a set X are the same as group homomorphisms from G to Sym ( X ). Suppose that a group G acts on a set X . Let Orb G ( x ) = { gx : g ∈ G} be the orbitof x ∈ X under the action of G , and Stab G ( x ) = { g ∈ G : gx = x } be the stabilizer of x underthe action of G . The Orbit-Stabilizer theorem states that the size of the orbit is the index of thestabilizer, that is | Orb G ( x ) | = [ G : Stab G ( x )]. We also recall that different orbits of the action aredisjoint and form a partition of X i.e., X = S { Orb G ( x ) : x ∈ X } . An alternating group is thegroup of even permutations of a finite set. Let { G i : i ∈ I } be an indexed collection of groups. De-fine Q weaki ∈ I G i = (cid:8) f : I → S i ∈ I G i (cid:12)(cid:12) ( ∀ i ∈ I ) f ( i ) ∈ G i , f ( i ) = 1 G i except f initely many i (cid:9) . The weak direct product of the groups { G i : i ∈ I } is the set Q weaki ∈ I G i with the operation of compo-nent wise multiplicative defined for all f, g ∈ Q weaki ∈ I G i by ( f g )( i ) = f ( i ) g ( i ) for all i ∈ I . AXIMAL INDEPENDENT SETS, VARIANTS OF CAC, AND CWF 5
Fraenkel–Mostowski permutation models.
We start with a ground model M of ZF A + AC where A is a set of atoms. Each permutation of A extends uniquely to a permutation of M by ǫ -induction. A permutation model N of ZFA is determined by a group G of permutationsof A and a normal filter F of subgroups of G . Let G be a group of permutations of A and F be a normal filter of subgroups of G . For x ∈ M , we denote the symmetric group with respectto G by sym G ( x ) = { g ∈ G | g ( x ) = x } . We say x is F -symmetric if sym G ( x ) ∈ F and x is hereditarily F -symmetric if x is F -symmetric and each element of transitive closure of x issymmetric. We define the permutation model N with respect to G and F , to be the class ofall hereditarily F -symmetric sets and recall that N is a model of ZF A (c.f. [[Jec73],
Theorem4.1 ]). If
I ⊆ P ( A ) is a normal ideal, then the filter base { fix G E : E ∈ I} generates a normalfilter over G , where fix G E denotes the subgroup { φ ∈ G : ∀ y ∈ E ( φ ( y ) = y ) } of G . Let I be anormal ideal generating a normal filter F I over G . Let N be the permutation model determinedby M, G , and F I . We say E ∈ I supports a set σ ∈ N if fix G E ⊆ sym G ( σ ). Lemma 2.5.
The following hold.(1) In every Fraenkel–Mostowski permutation model, CS implies vDCP (c.f. [[THS16],
The-orem 3.15(3) ]).(2) In ZFA, CWF implies LW (c.f. [[Tac18],
Lemma 5 ]).
Lemma 2.6. (c.f. [ [HT13] , Lemma 4.3 ]). Assume P is a set of prime numbers, M isa Fraenkel-Mostowski permutation model determined by the set A of atoms, the group G ofpermutations of A , and the filter F of subgroups of G . Assume further that(1) G is Abelian.(2) For every x ∈ M , Orb G ( x ) is finite.(3) There is a group G ∈ F such that for all φ ∈ G , if p is a prime divisor of the order of φ then p ∈ P .Then for every set Z ∈ M of non-empty sets there is a function f with domain Z such that forall y ∈ Z , ∅ 6⊆ f ( y ) ⊆ y and every prime divisor of | f ( y ) | is in P . Loeb’s theorem.
A topological space (
X, τ ) is called compact if for every U ⊆ τ such that S U = X there is a finite subset V ⊆ U such that S V = X . Lemma 2.7. (c.f. [ [Loeb65] , Theorem 1 ]). Let { X i } i ∈ I be a family of compact spaces whichis indexed by a set I on which there is a well-ordering ≤ . If I is an infinite set and there is achoice function F on the collection { C : C is closed, C = ∅ , C ⊂ X i for some i ∈ I } , then theproduct space Q i ∈ I X i is compact in the product topology. A theorem of Fulkerson and Gross.
Fulkerson–Gross [FG65] proved the followinglemma.
Lemma 2.8. (c.f. [ [Kom15] , Lemma 1 ], [FG65] ). A finite graph (
V, X ) is chordal if and onlyif there is an ordering < of V such that { w < v : { w, v } ∈ X } is a clique for each v ∈ V .3. Graph theoretical observations
Maximal independent set.Observation 3.1. (ZF) For every n ∈ ω \{ , } , P n is equivalent to AC n . Proof. ( ⇐ ) Fix n ∈ ω \{ , } , and let us assume AC n . Let G = ( V G , E G ) be a graph from theclass P K n (c.f. § , for definition of P K n ). Let { G i } i ∈ I = { ( V G i , E G i ) } i ∈ I be the componentsof G . By AC n select g i ∈ V G i for each i ∈ I . We can see that J = { g i : i ∈ I } is a maximalindependent set of G . For any g i , g j ∈ J such that g i = g j , we have { g i , g j } 6∈ E G . Consequently, J is an independent set. For the sake of contradiction, suppose J is not a maximal independentset. Then there is an independent set L which must contain two vertices x and y from V G i forsome i ∈ I . Since { x, y } ∈ E G , we obtain a contradiction. AMITAYU BANERJEE ( ⇒ ) Fix n ∈ ω \{ , } , and let us assume P n . Consider a system of n -element sets A = { A i } i ∈ I .We construct a graph G = ( V G , E G ). Constructing G : Let V G consists of all the pairs ( Y, y ) such that Y ∈ A and y ∈ Y , and theedge set is defined as follows { ( Y , y ) , ( Y , y ) } ∈ E G if and only if Y = Y and y = y .Clearly, the components of G are K n . By P n , G has a maximal independent set M . Since M isan independent set, for each Y ∈ A there is at most one y ∈ Y such that ( Y, y ) ∈ M . Since M is a maximal independent set, there is at least one y ∈ Y such that ( Y, y ) ∈ M . Consequently, M determines a choice function for A . (cid:3) Observation 3.2. (ZF) AC ωfin is equivalent to P lf,c . Proof. ( ⇒ ) We assume AC ωfin . Let G = ( V G , E G ) be some non-empty locally finite, connectedgraph. Consider some r ∈ V G . Let V = { r } . For each integer n ≥
1, define V n = { v ∈ V G : d G ( r, v ) = n } where ‘ d G ( r, v ) = n ’ means there are n edges in the shortest path joining r and v . Each V n is finite by locally finiteness of G , and V G = S n ∈ ω V n by connectedness of G . By U T ( ℵ , f in, ℵ ) (which is equivalent to AC ωfin (c.f. Definition 2.4 )), V G is countable.Consequently, V G is well-ordered. We prove that every graph based on a well-ordered set ofvertices has a maximal independent set in ZF. Let G = ( V G , E G ) be a graph on a well-orderedset of vertices V G = { v α : α < λ } . Thus we can use transfinite recursion, without using any formof choice, to construct a maximal independent set. Let M = ∅ . Clearly, M is an independentset. For any ordinal α , if M α is a maximal independent set, then we are done. Otherwise,there is some v ∈ V G \ M α , where M α ∪ { v } is an independent set of vertices. In that case, let M α +1 = M α ∪ { v } . For limit ordinals α , we use M α = S i ∈ α M i . Clearly, M = S i ∈ λ M i is amaximal independent set.( ⇐ ) We assume P lf,c . Since AC ωfin is equivalent to its partial version P AC ωfin (c.f.
Definition2.4 or [HR98]), it suffices to show
P AC ωfin . Let A = { A n : n ∈ ω } be a denumerable set ofnon-empty finite sets. Without loss of generality, we assume that A is disjoint. Consider adenumerable sequence T = { t n : n ∈ ω } disjoint from A . We construct a graph G = ( V G , E G ). • • • ... A • t • • • ... A • t ... ... Figure 1.
The graph G . Constructing G : Let V G = ( S n ∈ ω A n ) ∪ T . For each n ∈ ω , let { t n , t n +1 } ∈ E G and { t n , x } ∈ E G for every element x ∈ A n . Also for each n ∈ ω , and any two x, y ∈ A n such that x = y , let { x, y } ∈ E G (see Figure 1).Clearly, the graph G is connected and locally finite. By assumption, G has a maximal indepen-dent set of vertices, say M . Since M is maximal, M has to be infinite. Moreover, for each i ∈ ω ,either t i ∈ M or some v ∈ A i is in M . Since M is an independent set, for each i ∈ ω there isat most one v ∈ A i such that v ∈ M . Define M ′ = { v ∈ M : v ∈ A i for some i ∈ ω } . If M ′ isinfinite, then M ′ determines a partial choice function for A . Case (1).
Suppose M \ M ′ is finite. Then M ′ is infinite. Case (2).
Suppose M \ M ′ is infinite. Since { t n , t n +1 } ∈ E G for any n ∈ ω , if t n ∈ M \ M ′ , then t n +1 ∈ M ′ . Consequently, M ′ must be infinite as well. (cid:3) AXIMAL INDEPENDENT SETS, VARIANTS OF CAC, AND CWF 7
Observation 3.3. (ZF)
U T ( ℵ , ℵ , ℵ ) implies P lc,c , and P lc,c implies AC ℵ ℵ .Proof. In order to prove the first implication, let G = ( V G , E G ) be some non-empty locallycountable connected graph. Consider some r ∈ V G . Let V = { r } . For each integer n ≥ V n = { v ∈ V G : d G ( r, v ) = n } . Since G is locally countable, each V n is countable by U T ( ℵ , ℵ , ℵ ). Also V G = S n ∈ ω V n since G is connected. By U T ( ℵ , ℵ , ℵ ), V G is countable.Rest follows from the fact that every graph based on a well-ordered set of vertices has a maximalindependent set in ZF (c.f. the proof of Observation 3.2 ). The second assertion follows fromthe arguments of
Observation 3.2 , since AC ℵ ℵ is equivalent to P AC ℵ ℵ in ZF (c.f. Definition2.4 or [HR98]). (cid:3)
Remark 3.4.
Fix n ∈ ω \{ , } . We denote by C n the cycle graph with n -vertices. We denote by P C n , the class of those graphs whose only components are C n . We denote by P ′ n the statement ‘Every graph from the class P C n , has a maximal independent set’ . We remark that AC P n implies P ′ n in ZF where P n is the Perrin number of n . Let G = ( V G , E G ) be a graph from the class P C n . Let { G i } i ∈ I = { ( V G i , E G i ) } i ∈ I be the components of P C n . Let M i be the collection ofdifferent maximal independent sets of G i for each i ∈ I . Since the number of different maximalindependent sets in each component is P n , by AC P n we can choose a m i ∈ M i for each i ∈ I .Clearly, S i ∈ I m i is a maximal independent set of G .3.2. Locally finite connected graphs.Observation 3.5. (ZF) AC ωfin implies the statement ‘If ( V, X ) is a connected locally finitechordal graph, then there is an ordering < of V such that { w < v : { w, v } ∈ X } is a clique foreach v ∈ V ’.Proof. We note that by arguments in the proof of
Observation 3.2 , it is enough to see thatthe statement ‘If ( V, X ) is a chordal graph based on a well orderable set of vertices, then there isan ordering < of V such that { w < v : { w, v } ∈ X } is a clique for each v ∈ V ’ is provable in ZF.By Lemma 2.8 , each finite subgraph (
W, X | W ) has an ordering such that { w < v : { w, v } ∈ X ↾ W } is a clique for every v ∈ W . We can encode every total ordering of a set W by a choiceof one of <, = , > for each pair ( x, y ) ∈ W × W . Endow { <, = , > } with the discrete topology and T = { <, = , > } V × V with the product topology. Since V is well-ordered, V × V is well-ordered inZF. Consequently, { <, = , > } × { V × V } is well-ordered in ZF. By Lemma 2.7 , T is compact.We use the compactness of T to prove the existence of the desired ordering. (cid:3) Remark 3.6.
We list some other graph-theoretical statements from different papers, restrictedto locally finite connected graphs, which are related to AC ωfin .(1) Komj´ath–Galvin [KG91] proved that any graph based on a well-ordered set of verticeshas a chromatic number and an irreducible good coloring in ZF. Consequently, thestatements ‘any locally finite connected graph has a chromatic number’ and ‘any locallyfinite connected graph has an irreducible good coloring’ are provable under AC ωfin in ZF.(2) Hajnal [[Haj85], Theorem 2 ] proved that if the chromatic number of a graph G isfinite (say k < ω ), and the chromatic number of another graph G is infinite, then thechromatic number of G × G is k . In [BG20] we observed that if G is based on awell-ordered set of vertices, then the following statement holds in ZF. ‘ χ ( E G ) = k < ω and χ ( E G ) ≥ ω implies χ ( E G × G ) = k .’ Consequently, under AC ωfin the above statement holds in ZF if G is a locally finiteconnected graph.(3) Delhomm´e and Morillon [DM06] proved that AC ωfin is equivalent to the statement ‘Everylocally finite connected graph has a spanning tree’ in ZF. We use the fact that the number of different maximal independent sets in an n-vertex cycle graph is the n-thPerrin number for 1 < n < ω . AMITAYU BANERJEE A variant of CAC
Tachtsis communicated to us the following lemma.
Lemma 4.1.
The following holds.(1)
U T ( ℵ , ℵ , ℵ ) implies the statement ‘If ( P, ≤ ) is a poset such that P is well-ordered, andif all antichains in P are finite and all chains in P are countable, then P is countable’ .(2) ℵ is regular implies the statement ‘If ( P, ≤ ) is a poset such that P is well-ordered, andif all antichains in P are finite and all chains in P are countable, then P is countable’ . Proof.
We prove (1). Let ( P, ≤ ) be a poset such that P is well-ordered, all antichains in P are finite, and all chains are countable. Fix a well-ordering (cid:22) of P . By way of contradiction,assume that P is uncountable. We construct an infinite antichain to obtain a contradiction.Since P is well-ordered by (cid:22) , we may construct (via transfinite induction) a maximal ≤ -chain, V say, without invoking any form of choice. Since V is countable, it follows that P − V is uncountable and every element of P − V is incomparable to some element of V . Thus P − V = S { W p : p ∈ V } , where W p is the set of all elements of P − V which are incomparableto p . Since P − V is uncountable and V is countable, it follows by U T ( ℵ , ℵ , ℵ ) that W p is uncountable for some p in V . Let p be the least (with respect to (cid:22) ) such element of V .Now, construct a maximal ≤ -chain in (the uncountable set) W p , V say, and let (similarly tothe above argument) p be the least (with respect to (cid:22) ) element of V such that the set W p of all elements of W p which are incomparable to p is uncountable. Continuing in this fashionby induction (and noting that the process cannot stop at a finite stage), we obtain a countablyinfinite antichain { p n : n ∈ ω } , contradicting the assumption that all antichains are finite.Therefore, P is countable.Similarly, we can prove (2). (cid:3) Modifying
Lemma 4.1 , we may observe that
U T ( ℵ α , ℵ α , ℵ α ) implies the statement ‘If ( P, ≤ ) is a poset such that P is well-ordered, and if all antichains in P are finite and all chains in Phave size ℵ α , then P has size ℵ α ’ for any regular ℵ α in ZF. Corollary 4.2.
The statement ‘If ( P, ≤ ) is a poset such that P is well-ordered, and if allantichains in P are finite and all chains in P are countable, then P is countable’ holds in anyFraenkel-Mostowski model. Proof.
Follows from the fact that the statement ℵ is a regular cardinal holds in every Fraenkel-Mostowski model (c.f. [[HKRST01], Corollary 1 ]). (cid:3)
Theorem 4.3. (ZFA)
For any regular ℵ α , and n ∈ ω \{ , } , CAC ℵ α does not imply AC − n .Proof. Halbeisen–Tachtsis [[HT20],
Theorem 8 ] constructed a permutation model (we donoteby N HT ( n )) where for arbitrary n ≥ AC − n fails but CAC holds. We fix an arbitrary integer n ≥ Theorem 8 ] as follows.
Defining the ground model M : We start with a ground model M of ZF A + AC where A isa countably infinite set of atoms written as a disjoint union S { A i : i ∈ ω } where for each i ∈ ω , A i = { a i , a i , ...a i n } . Defining the group G and the filter F of subgroups of G : • Defining G : G is defined in [HT20] in a way so that if η ∈ G , then η only movesfinitely many atoms and for all i ∈ ω , η ( A i ) = A k for some k ∈ ω . We recall the detailsfrom [HT20] as follows. For all i ∈ ω , let τ i be the n -cycle a i a i ...a i n a i .For every permutation ψ of ω , which moves only finitely many natural numbers, let φ ψ be the permutation of A defined by φ ψ ( a i j ) = a ψ ( i ) j for all i ∈ ω and j = 1 , , ..., n . Let η ∈ G if and only if η = ρφ ψ where ψ is a permutation of ω which moves only finitelymany natural numbers and ρ is a permutation of A for which there is a finite F ⊆ ω AXIMAL INDEPENDENT SETS, VARIANTS OF CAC, AND CWF 9 such that for every k ∈ F , ρ ↾ A k = τ jk for some j < n , and ρ fixes A m pointwise forevery m ∈ ω \ F . • Defining F : Let F be the filter of subgroups of G generated by { fix G ( E ) : E ∈ [ A ] <ω } . Defining the permutation model:
Consider the FM-model N HT ( n ) determined by M , G and F .Following point 1 in the proof of [[HT20], Theorem 8 ], both A and A = { A i } i ∈ ω are amor-phous in N HT ( n ) and no infinite subfamily B of A has a Kinna–Wegner selection function.Consequently, AC − n fails. We follow the steps below to prove that for any regular ℵ α , CAC ℵ α holds in N HT ( n ).(1) Let ( P, ≤ ) be a poset in N HT ( n ) such that all antichains in P are finite and all chainsin P have size ℵ α . Let E ∈ [ A ] <ω be a support of ( P, ≤ ). We can write P as a disjointunion of x G ( E )-orbits, i.e., P = S { Orb E ( p ) : p ∈ P } , where Orb E ( p ) = { φ ( p ) : φ ∈ x G ( E ) } for all p ∈ P . The family { Orb E ( p ) : p ∈ P } is well-orderable in N HT ( n ) sincex G ( E ) ⊆ Sym G ( Orb E ( p )) for all p ∈ P .(2) Since if η ∈ G , then η only moves finitely many atoms , Orb E ( p ) is an antichain in P for each p ∈ P . Otherwise there is a p ∈ P , such that Orb E ( p ) is not an antichain in( P, ≤ ). Thus, for some φ, ψ ∈ fix G ( E ), φ ( p ) and ψ ( p ) are comparable. Without loss ofgenerality we may assume φ ( p ) < ψ ( p ). Since if η ∈ G , then η only moves finitelymany atoms , there exists some k < ω such that φ k = 1 A . Let π = ψ − φ . Consequently, π ( p ) < p and π k = 1 A for some k ∈ ω . Thus, p = π k ( p ) < π k − ( p ) < ... < π ( p ) < p . Bytransitivity of < , p < p , which is a contradiction.(3) Since Orb E ( p ) is an antichain, it is finite. Consequently, Orb E ( p ) is well-orderable.Since U T ( W O, W O, W O ) holds in N HT ( n ), P is well-orderable by (1) and (2). Also wenote that U T ( W O, W O, W O ) implies UT( ℵ α , ℵ α , ℵ α ) in any FM-model (c.f. page 176of [HR98]). So, we are done by Lemma 4.1 and the point noted in the paragraph after
Lemma 4.1 . (cid:3) Theorem 4.4. (ZFA)
For any regular ℵ α , CAC ℵ α does not imply ‘There are no amorphoussets’.Proof. We consider the basic Fraenkel model (labeled as Model N in [HR98]) where ‘there areno amorphous sets’ is false, and U T ( W O, W O, W O ) holds (c.f. [HR98]). Let ( P, ≤ ) be a posetin N , and E be a nite support of ( P, ≤ ). By the arguments of the proof of Theorem 4.3 , O = { Orb E ( p ) : p ∈ P } is a well-ordered partition of P . Now for each p ∈ P , Orb E ( p ) isan antichain (c.f. the proof of [[Jec73], Lemma 9.3 ]). Thus, by methods from the proof of
Theorem 4.3 , CAC ℵ α holds in N . (cid:3) Remark 4.5.
Since
U T ( W O, W O, W O ) holds in N HT ( n ) and N , AC ωfin holds in N HT ( n ) and N . Consequently, by Observation 3.2 , P lf,c holds in N HT ( n ) and N . Theorem 4.6. (ZF)
CAC ℵ implies P AC ℵ fin .Proof. Let A = { A n : n ∈ ℵ } be a family of non-empty nite sets. Without loss of generality,we assume that A is disjoint. Dene a binary relation ≤ on A = S A as follows: for all a, b ∈ A ,let a ≤ b if and only if a = b or a ∈ A n and b ∈ A m and n < m . Clearly, ≤ is a partial orderon A . Also, A is uncountable. The only antichains of ( A, ≤ ) are the nite sets A n where n ∈ ℵ .By CAC ℵ , A has an uncountable chain, say C . Let M = { m ∈ ℵ : C ∩ A m = ∅} . Since C isa chain and A is the family of all antichains of ( A, ≤ ), we have M = { m ∈ ℵ : | C ∩ A m | = 1 } .Clearly, f = { ( m, c m ) : m ∈ M } , where for m ∈ M , c m is the unique element of C ∩ A m , isa choice function of the uncountable subset B = { A m : m ∈ M } of A . Thus B is a ℵ -sizedsubfamily of A with a choice function. (cid:3) Theorem 4.7. (ZFA) DC does not imply
CAC ℵ . Proof.
We recall Jech’s model (labeled as N ( ℵ α ) in [HR98]). • Defining the ground model M . We start with a ground model M of ZF A + AC withan ℵ α -sized set A of atoms which is a disjoint union of ℵ α pairs, so that A = S { A γ : γ < ℵ α } , A γ = { a γ , b γ } . • Defining the group G of permutations and the filter F of subgroups of G .– Defining G . Let G be the group of all permutations of A which fix A γ for all γ < ℵ α . – Defining F . Let F be the normal filter on G which is generated by { fix G ( E ) : E ⊂ A , | E | < ℵ α } . • Defining the permutation model.
Consider the permutation model N ( ℵ α ) deter-mined by M , G and F .Jech proved that DC < ℵ α is true in N ( ℵ α ). Let us consider the model N ( ℵ ). Clearly, DC < ℵ is true in N ( ℵ ). By Theorem 4.6 , it is enough to show that
P AC ℵ fin fails in the model.We prove that the family A = { A γ : γ < ℵ } of finite sets has no subfamily B of cardinality ℵ , such that B has a choice function. For the sake of contradiction, let B be a subfamily ofcardinality ℵ of A with a choice function f ∈ N ( ℵ ) and support E ∈ [ A ] < ℵ . Since E iscountable, there is a γ < ℵ such that A γ ∈ B and A γ ∩ E = ∅ . Without loss of generality, let f ( A γ ) = a γ . Consider the permutation π which is the identity on A η , for all η ∈ ℵ \{ γ } , andlet ( π ↾ A γ )( a γ ) = b γ = a γ . Then π fixes E pointwise, hence π ( f ) = f . So, f ( A γ ) = b γ whichcontradicts the fact that f is a function. (cid:3) Cofinal well-founded subsets in ZFA
We modify the arguments from [[THS16],
Theorem 3.26 ] and [[Tac18],
Theorem 10(ii) ] toobserve the following.
Lemma 5.1.
Let A be a set of atoms. Let G be the group of permutations of A such that eithereach η ∈ G moves only finitely many atoms or there is a n ∈ ω \{ , } , such that for all η ∈ G , η n = 1 A . Let F be the normal filter of subgroups of G generated by { fix G ( E ) : E ∈ [ A ] <ω } .Then in the Fraenkel-Mostowski model N determined by A , G , and F , CS and CWF hold.Consequently, vDCP and LW hold.Proof. We follow the steps below.(1) Let ( P, ≤ ) be a poset in N and E ∈ [ A ] <ω be a support of ( P, ≤ ). We can write P as adisjoint union of x G ( E )-orbits, i.e., P = S { Orb E ( p ) : p ∈ P } , where Orb E ( p ) = { φ ( p ) : φ ∈ x G ( E ) } for all p ∈ P . The family { Orb E ( p ) : p ∈ P } is well-orderable in N sincex G ( E ) ⊆ Sym G ( Orb E ( p )) for all p ∈ P .(2) We prove that Orb E ( p ) is an antichain in P for each p ∈ P . Otherwise there is a p ∈ P ,such that Orb E ( p ) is not an antichain in ( P, ≤ ). Thus, for some φ, ψ ∈ fix G ( E ), φ ( p )and ψ ( p ) are comparable. Without loss of generality we may assume φ ( p ) < ψ ( p ). Let π = ψ − φ . Consequently, π ( p ) < p . Case 1:
Suppose there is a n ∈ ω \{ , } , such that for every η ∈ G , η n = 1 A . So π n = 1 A . Thus, p = π n ( p ) < π n − ( p ) < ... < π ( p ) < p . By transitivity of < , p < p ,which is a contradiction. Case 2:
Suppose each η ∈ G , moves only finitely many atoms. Then for some k < ω , π k = 1. Rest follows from the arguments in Case 1 .(3) We can follow [[THS16],
Theorem 3.26 ] to see that CS holds in N .(4) Although in every Fraenkel-Mostowski model, CS implies vDCP in ZFA (c.f. Lemma2.5 ), we can recall the arguments from the 1 st -paragraph of [[THS16], Page175 ] to givea direct proof of vDCP in N .(5) We can follow [[Tac18], Theorem 10 (ii) ] to see that CWF holds in N . By Lemma2.5 , LW holds in N . AXIMAL INDEPENDENT SETS, VARIANTS OF CAC, AND CWF 11 (cid:3)
A model of ZFA.
Herrlich, Howard, and Tachtsis [[HHT12],
Theorem 11 , Case 1 , Case2 ] constructed two different classes of permutation models. Halbeisen–Tachtsis [[HT20],
Theo-rem 10(ii) ] proved that LOC − does not imply LOKW − in ZFA. For the sake of convenience,we denote by N HT , the permutation model of [[HT20], Theorem 10(ii) ]. The model N HT isvery similar to the model from [[HHT12], Theorem 11 , Case 2 ] except the fact that in N HT each permutation φ in the group G of permutations of the sets of atoms, can move only finitelymany atoms. Fix a natural number n such that n = 3 or n >
4. We construct a model M n ofZFA similar to the model constructed in [[HHT12], Theorem 11 , Case 1 ], where each permuta-tion φ in the group G of permutations of the sets of atoms, can move only finitely many atoms.Consequently, by Lemma 5.1 , CS, vDCP, CWF, and LW hold in M n . In particular we provethat (LOC − + CS + CWF) does not imply LOC − n in ZFA if n ∈ ω such that n = 3 or n > Theorem 5.2.
Let n be a natural number such that n = 3 or n > . Then there is a model M n of ZFA where the following hold.(1) If X ∈ { LOC − , CS, vDCP, CW F, LW } , then X holds.(2) LOC − n fails.(3) If X ∈ {P n , DT, LT } , then X fails.Proof. Fix a natural number n such that n = 3 or n > Defining the ground model M : Let κ be any infinite well-ordered cardinal number. We startwith a ground model M of ZF A + AC where A is a κ -sized set of atoms written as a disjointunion S { A α : α < κ } , where A α = { a α, , a α, , ..., a α,n } such that | A α | = n for all α < κ . Defining the group G and the filter F of subgroups of G : • Defining G : Let G be the weak direct product of G α ’s where G α is the alternating groupon A α for each α < κ . Hence, a permutation η of A is an element of G if and only if forevery α < κ , η ↾ A α ∈ G α , and η ↾ A α = 1 A α for all but nitely many ordinals α < κ .Consequently, every element η ∈ G moves only nitely many atoms . • Defining F : Let F be the normal filter of subgroups of G generated by { fix G ( E ) : E ∈ [ A ] <ω } . Defining the permutation model:
Consider the permutation model M n determined by M , G and F . (1). If X ∈ { LOC − , CS, vDCP, CW F, LW } , then X holds in M n : Since every permutation φ ∈ G moves only finitely many atoms, CS, vDCP, CWF, and LW holds in M n by Lemma5.1 . Applying the group-theoretic facts from [[HHT12],
Theorem 11 , Case 1 ] and followingthe arguments of the proof of [[HT20],
Theorem 10(ii) ] we may observe that LOC − holds in M n . (2). LOC − n fails in M n : We prove that in M n , the well-ordered family A = { A α : α < κ } of n element sets does not have a partial choice function. For the sake of contradiction, let B be an innite subfamily of A with a choice function f ∈ M n and support E ∈ [ A ] <ω . Since E is finite, there is an i < κ such that A i ∈ B and A i ∩ E = ∅ . Without loss of generality, let f ( A i ) = a i . Consider the permutation π which is the identity on A j , for all j ∈ κ − i , and let( π ↾ A i )( a i ) = a i = a i . Then π fixes E pointwise, hence π ( f ) = f . So, f ( A i ) = a i whichcontradicts the fact that f is a function. Thus LOC − n fails in M n . (3). If X ∈ {P n , DT, LT } , then X fails in M n : Since AC n fails in the model from the ar-guments of the previous paragraph, P n fails in the model by Observation 3.1 . Since in M n ,the linearly-ordered family A = { A α : α < κ } of n element sets does not have a choice function,DT fails in M n by [[Tac19], Theorem 3.1(ii) ]. Since in every Fraenkel–Mostowski model ofZFA, LT implies AC W O (c.f.[[Tac19a],
Theorem 4.6(i) ]), LT fails in M n since the well-orderedfamily A = { A α : α < κ } does not have a choice function. (cid:3) Corollary 5.3. (ZFA) (LOC − + CS + CWF) does not imply CAC ℵ .Proof. Consider the permutation model M n constructed in Theorem 5.2 by letting the infinitewell-ordered cardinal number κ to be ℵ . Rest follows from Theorem 4.6 and the argumentsof
Theorem 5.2(2) . (cid:3) Following the arguments in the proof of
Theorem 5.2(3) , we can also observe that DT and LTfails in the model from [[HT20],
Theorem 10(ii) ].5.2.
The model N ∗ ( p ) . We improve the choice strength of the result of [[CHHKR08],
Theorem4.8 ] if k is a prime, applying the methods of [HT13]. Theorem 5.4.
Fix a prime p ∈ ω \{ , , } . Then in N ∗ ( p ) , the following hold.(1) CWF holds.(2) M C ( q ) holds for all prime q = p .(3) If p is not a divisor of n , then AC n and P n hold.Proof. (1). CWF holds: We note that N ∗ ( p ) was constructed via a group G such that G wasabelian and for all φ ∈ G , φ p = 1 A (c.f. [[CHHKR08], Theorem 4.8 ]). Also the normal filter F of subgroups of G was generated by { fix G ( E ) : E ∈ [ A ] <ω } . Thus, CWF holds in N ∗ ( p ) by Lemma 5.1 . (2). M C ( q ) holds for all prime q = p : We prove that in N ∗ ( p ), M C ( q ) holds for all prime q = p . To see this we observe that N ∗ ( p ) satisfies the hypotheses of Lemma 2.6 with P = { p } . • First, we note that G is Abelian (c.f. [[CHHKR08], Theorem 4.8 ]). • We follow the arguments from the proof of [[HT13],
Theorem 4.6 ] to see that forall t ∈ N ∗ ( p ), Orb G ( t ) is finite. Fix a t ∈ N ∗ ( p ). By the Orbit-Stabilizer theorem, | Orb G ( t ) | = [ G /Stab G ( t )], where Stab G ( t ) is the stabilizer subgroup of G with respect to t , i.e., Stab G ( t ) = { g ∈ G : g ( t ) = t } . Let E t = ∪ li =0 A i be a support of t . Clearly, if φ, ψ ∈ G which agree on E t , then φStab G ( t ) = ψStab G ( t ). By the definition of G , for all φ ∈ G , φ p = 1 A . So [ G /Stab G ( t )] ≤ p l +1 . Thus Orb G ( t ) is finite. • Since G is such that for all φ ∈ G , φ p = 1 A (c.f. [[CHHKR08], Theorem 4.8 ]), we cansee that part (3) of
Lemma 2.6 is also satisfied. Fix ψ ∈ G . Let p be a prime divisorof the order of ψ (i.e., p ). Clearly, p = p ∈ P .By Lemma 2.6 , for every family { X i : i ∈ I } of non-empty sets in N ∗ ( p ), there is a function F with domain I such that for all i ∈ I , we have that F ( i ) ⊆ X i and for all i ∈ I , every primedivisor of | F ( i ) | is in P . Thus for every prime q = p , M C ( q ) is true. (3). If p is not a divisor of n , then AC n and P n hold: If p is not a divisor of n , then AC n holds, by the arguments in the proof of [[HT13], Theorem 4.7 (2)]. Consequently, if p is not adivisor of n , then P n holds by Observation 3.1 . (cid:3) Remark 5.5.
We observe that CWF holds in the Second Fraenkel’s model (labeled as Model N in [HR98]). Moreover, if X ∈ {P lf,c , P } , then X fails in N . • We note that N was constructed via a group G such that for all φ ∈ G , φ = 1 A . By Lemma 5.1 , CWF holds in N . • Since AC fails in N (c.f. [HR98]), P fails in N by Observation 3.1 . • Since AC ωfin fails in N (c.f. [HR98]), P lf,c fails in N by Observation 3.2 . Remark 5.6.
Fix a prime p ∈ ω . Howard–Tachtsis [[HT13], Theorem 4.7 ] proved that
M C ( q ) holds in N ( p ) (c.f. the model from [[HT13], § ]) for every prime q = p . Fix aprime p ∈ ω \{ , , } . • We note that N ( p ) was constructed via a group G such that for all φ ∈ G , φ p = 1 A .Consequently, by Lemma 5.1 , CWF holds in N ( p ). AXIMAL INDEPENDENT SETS, VARIANTS OF CAC, AND CWF 13 • In N ( p ), AC n is true for all n ∈ ω \{ , } such that p is not a divisor of n (c.f. [[HT13], Theorem 4.7(2) ]). Consequently, by
Observation 3.1 , P n holds in N ( p ) if p is nota divisor of n . • Since AC ωfin fails in N ( p ) (c.f. the proof of [[HT13], Theorem 4.7(3) ]), P lf,c fails in N ( p ) by Observation 3.2 . 6.
Summary
Synopsis of theorems, observations, and remarks. • (ZF) ( ∀ n ∈ ω \{ , } ) AC n ↔ P n (c.f. [ § Observation 3.1 ]). • (ZF) U T ( ℵ , ℵ , ℵ ) → P lc,c → AC ℵ ℵ → AC ωfin ←→ P lf,c (c.f. [ § Observation 3.2 , Observation 3.3 ]). • (ZF) AC ωfin implies the statement ‘If G = ( V G , E G ) is a connected locally finite chordalgraph, then there is an ordering < of V G such that { w < v : { w, v } ∈ E G } is a clique foreach v ∈ V G ’ (c.f. [ § Observation 3.5 ]). • (ZFA) For every n ∈ ω \{ , } , for any regular ℵ α , CAC ℵ α AC − n (c.f. [ § Theorem4.3 ]). • (ZFA) For any regular ℵ α , CAC ℵ α ‘There are no amorphous sets’ (c.f. [ § Theorem4.4 ]). • In N HT ( n ) and N , P lf,c holds (c.f. [ § Remark 4.5 ]). • (ZF) CAC ℵ → P AC ℵ fin (c.f. [ § Theorem 4.6 ]). • (ZFA) DC CAC ℵ (c.f. [ § Theorem 4.7 ]). • (ZFA) Let n ∈ ω such that n = 3 or n >
4. Then (LOC − + CS + CWF) X , if X ∈ { LOC − n , DT, LT } (c.f. [ § Theorem 5.2 ]). • (ZFA) (LOC − + CS + CWF) CAC ℵ (c.f. [ § Corollary 5.3 ]). • For any prime p ∈ ω \{ , , } , CWF holds in N ∗ ( p ), N , and, N ( p ). • For any prime p ∈ ω \{ , , } , if p is not a divisor of n , then AC n and P n hold in N ∗ ( p ).(c.f. [ § Theorem 5.4 ]). • If X ∈ {P lf,c , P } , then X fails in N (c.f. [ § Remark 5.5 ]). • For any prime p ∈ ω \{ , , } , P n holds in N ( p ) if p is not a divisor of n , and P lf,c fails in N ( p ) (c.f. [ § Remark 5.6 ]).6.2.
Table of statements and models.
The following table depicts the truth/falsity of state-ments that we studied in different permutation models. The bold-letter entries ‘ T (True) and‘ F ’ (False) denote the new results in this note. The normal-letter F and T denote the knownresults.Table of statements depicting their truth/falsity in certain modelsModels CAC ℵ α CWF P lf,c P n N ∗ ( p ) ( p > T F if p = 3 T if p n N (Remark 5.5) T F F if n = 2 N ( p ) ( p ∈ ω \{ , , } )(Remark 5.6) T F T if p n M n ( n = 3 /n > T F N HT ( n ) ( n ≥ T T T F N (Theorem 4.4) T T T F if n = 2 In [BG20], we observed that CWF holds in N HT ( n ). In N HT ( n ), AC n fails. Consequently, P n fails in the model by Observation 3.1 . Since AC ωfin fails in N ∗ (3) (see [HR98]), P lf,c fails in N ∗ (3) by Observation 3.3 . In N , AC fails (c.f. [HR98]). So P fails in N .7. Acknowledgements
The author would like to thank Eleftherios Tachtsis for communicating
Lemma 4.1 to us in aprivate conversation.
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Department of Logic, Institute of Philosophy, E¨otv¨os Lor´and University, M´uzeum krt. 4/i Budapest,H-1088 Hungary
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