Infinite Stable Graphs With Large Chromatic Number
aa r X i v : . [ m a t h . L O ] S e p INFINITE STABLE GRAPHS WITH LARGE CHROMATICNUMBER
YATIR HALEVI, ITAY KAPLAN, AND SAHARON SHELAH
Abstract.
We prove that if G = ( V, E ) is an ω -stable (respectively, super-stable) graph with χ ( G ) > ℵ (respectively, 2 ℵ ) then G contains all the finitesubgraphs of the shift graph Sh n ( ω ) for some n . We prove a variant of thistheorem for graphs interpretable in stationary stable theories. Furthermore, if G is ω -stable with U( G ) ≤ n ≤ Introduction
The chromatic number χ ( G ) of a graph G = ( V, E ) is the minimal cardinal κ forwhich the exists a vertex coloring with κ colors. There is a long history of struc-ture theorems deriving from large chromatic number assumptions. For example if χ ( G ) ≥ ℵ then G must contain all finite bipartite graphs [EH66, Corollary 5.6]and every sufficiently large odd circuit [EHS74, Theorem 3], [Tho83]. See [Kom11]for more information.In [Tay71, Problem 1.14], Taylor asked what is the least cardinal κ such thatevery graph G with χ ( G ) ≥ κ is elementary equivalent to graphs of arbitrarily largechromatic number. It is clear that such a minimal cardinal exists (see [Tay71, The-orem 1.13]). Taylor noted that necessarily κ ≥ ℵ . Nowadays, Taylor’s conjectureis usually phrased in the following way (see [Kom11, Section 3]). Conjecture (Taylor’s Conjecture) . For any graph G with χ ( G ) ≥ ℵ and cardinal κ there exists a graph H with χ ( H ) ≥ κ such that G and H share the same finitesubgraphs.For a caridnal κ the shift graph Sh n ( κ ) is the graph whose vertices are increas-ing n -tuples s of ordinals less than κ , where we put an edge between s and t iffor every 1 ≤ i ≤ n − s ( i ) = t ( i −
1) or vice-versa. The shift graphs Sh n ( κ )have large chromatic numbers depending on κ , see Fact 2.6. Erd¨os-Hajnal-Shelah[EHS74, Problem 2] and Taylor [Tay70, Problem 43, page 508] proposed the follow-ing strengthening of this conjecture. Conjecture (Strong Taylor’s Conjecture) . For any graph G with χ ( G ) ≥ ℵ thereexists an n ∈ N such that G contains all finite subgraphs of Sh n ( ω ).Assuming the strong Taylor’s Conjecture, if χ ( G ) ≥ ℵ there exists an elementaryextension G ≺ G that has Sh n ( i n − ( κ ) + ) as a subgraph, and thus χ ( G ) ≥ κ + ,see Fact 2.6. So the strong Taylor’s conjecture implies Taylor’s conjecture. It isknown that Taylor’s conjecture is consistently false and that a relaxation of Taylor’s Mathematics Subject Classification.
Key words and phrases. chromatic number; stable graphs; Taylor’s conjecture.The first author would like to thanks the Israel Science Foundation for its support of thisresearch (grant No. 181/16) and the Kreitman foundation fellowship. The second author wouldlike to thank the Israel Science Foundation for its support of this research (grants no. 1533/14and 1254/18). The third author would like to thank the Israel Science Foundation grant no:1838/19 and the European Research Council grant 338821. Paper no. 1196 in the third author’spublication list. conjecture is consistently true, namely assuming that χ ( G ) ≥ ℵ [KS05]. The strongTaylor’s conjecture was refuted in [HK84, Theorem 4].Since the (strong) Taylor’s conjecture fails in general, one may wonder if it holdsfor a “tame” class of graphs. Classification theory provides “dividing lines” sepa-rating “tame” and “wild” classes of structures (and theories). These dividing linesare usually defined by requiring that a structure omits a certain class of (definable)combinatorial patterns. It is thus not surprising that restricting to such graphs willyield better combinatorial results.An important instance of this phenomena is when tame=stable. Stable theories,which originated in the work of the third author in the 60s and 70s, is the mostextensively studied class. Examples of stable theories include abelian groups, mod-ules, algebraically closed fields, graph theoretic trees, or more generally superflatgraphs [PZ78]. Stablility also had an impact in combinatorics, e.g. [MS14] and[CPT20] to name a few.In this paper we prove variants of the strong Taylor’s conjecture for some classesof stable graphs. Theorem.
Let G = ( V, E ) be a graph. If(1) G is ω -stable and χ ( G ) > ℵ or(2) G is superstable and χ ( G ) > ℵ or(3) G is interpretable in a stable structure, in which every type (over any set)is stationary, and χ ( G ) > i ( ℵ ) then G contains all finite subgraphs of Sh n ( ω ) for some n ∈ N .Furthermore, if G is ω -stable with χ ( G ) > ℵ and U( G ) ≤ then n ≤ suffices. Items (1) and (2) are Corollary 4.4, (3) is Corollary 5.21 and the furthermore isTheorem 6.9.The following remains open.
Question . (1) What is the situation with general stable graphs?(2) Is it enough to assume χ ( G ) > ℵ in the above theorem?(3) What about other tameness assumptions, e.g. NIP? Acknowledgments.
Section 6 is joint work with Elad Levi. We would like tothank him for allowing us to add these results.2.
Notation and Preliminaries
We use fairly standard model theoretic terminology and notation, see for example[TZ12]. We use small latin letters a, b, c for tuples and capital letters
A, B, C forsets. We also employ the standard model theoretic abuse of notation and write a ∈ A even for tuples when the length of the tuple is immaterial or understoodfrom context. When we write a ≡ A b we mean that tp( a/A ) = tp( b/A ).For any two sets A and J , let A J be the set of injective functions from γ to A (where the notation is taken from the falling factorial notation), and if ( A, < ) and(
J, < ) are both linearly ordered sets, let ( A J ) < be the subset of A J consisting ofstrictly increasing functions. If we want to emphasize the order on J we will write( A ( J,< ) ) < . For an ordinal γ , we set A <γ := S α<γ A α . Throughout this paper,we interchangeably use sequence notation and function notation for elements of A J , e.g. for f ∈ A J , f ( i ) = f i . For any sequence η we denote by Range( η ) theunderlying set of the sequence (i.e. its image). If ( A, < A ) and ( B, < B ) are linearlyordered sets, then the most significant coordinate of the lexicographic order on A × B is the first one.By a graph we mean a pair G = ( V , E ) where E ⊆ V is symmetric andirreflexive. A graph homomorphism between G = ( V , E ) and G = ( V , E ) is a NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 3 map f : V → V such that f ( e ) ∈ E for every e ∈ E . If f is injective we will saythat f embeds G into G a subgraph. If in addition we require that f ( e ) ∈ E ifand only if e ∈ E we will say that f embeds G into G as an induced subgraph. Definition 2.1.
Let G = ( V, E ) be a graph.(1) For a cardinal κ , a vertex coloring (or just coloring) of size κ is a function c : V → κ such that x E y implies c ( x ) = c ( y ) for all x.y ∈ V .(2) The chromatic number χ ( G ) is the minimal cardinality of a vertex coloringof G . Remark . Note that for a graph G = ( V, E ) with | V | ≥ χ ( G ) = 1 if and onlyif | E | = ∅ .Here are some useful easy and well known properties of the chromatic numberfunction of graphs (we provide proofs for the convenience of the reader). Lemma 2.3.
Let G = ( V, E ) be a graph.(1) If V = S i ∈ I V i then χ ( G ) ≤ P i ∈ I χ ( V i , E ↾ V i ) .(2) If E = S i ∈ I E i (with the E i being symmetric) then χ ( G ) ≤ Q i ∈ I χ ( V, E i ) .(3) If ϕ : H → G is a graph homomorphism then χ ( H ) ≤ χ ( G ) .(4) If ϕ : ( H, E H ) → ( G, E G ) is a surjective graph homomorphism with e ∈ E H ⇐⇒ ϕ ( e ) ∈ E G then χ ( H ) = χ ( G ) .Proof. (1) Let c i : V i → κ i be a coloring of ( V i , E ↾ V i ). Define a coloring c : V → S { κ i × { i } : i ∈ I } by choosing for any v ∈ V an i v ∈ I such that v ∈ V i v andsetting c ( v ) = ( c i v ( v ) , i v ).(2) Let c i : V i → κ i be a coloring of ( V, E i ). Define a coloring c : V → Q i ∈ I κ i by c ( v )( i ) = c i ( v ).(3) Write G = ( V G , E G ) and H = ( V H , E H ) and let c : V G → κ be a coloringof G . Define a coloring c ′ : V H → κ of H by c ′ ( v ) = c ( f ( v )).(4) Let c : V H → κ be a coloring. We define a coloring c ′ : V G → κ by choosingfor any element v ∈ V G an element w ∈ ϕ − ( v ) and setting c ′ ( v ) = c ( w ). Itis a legal coloring since if v E G v then w E H w for any w ∈ ϕ − ( v ) and w ∈ ϕ − ( v ). (cid:3) We will mainly be interested with the following so called “Shift Graphs”, firstdefined by Erd¨os-Hajnal in [EH68].
Example 2.4 (Shift Graph) . For any finite number 1 ≤ r and any linearly orderedset ( A, < ), let Sh r ( A ), or Sh r ( A, < ) if we want to emphasize the order, (the shiftgraph on A ) be the following graph: its set of vertices is the set ( A r ) < of increasing r -tuples, s , . . . , s r − , and we put an edge between s and t if for every 1 ≤ i ≤ r − s ( i ) = t ( i − r ( A ) is a connectedgraph. If r = 1 this gives K A , the complete graph on A . Example 2.5 (Symmetric Shift Graph) . Let 1 ≤ r be any natural number and A any set. The symmetric shift graph Sh symr ( A ) is defined similarly as the shiftgraph but with set of vertices A r (set of distinct r -tuples). Note that Sh r ( A ) isan induced subgraph of Sh symr ( A ) (and that for r = 1 they are both the completegraph on A ). Recall that i ( κ ) := κ and i k +1 ( κ ) := 2 i k ( κ ) . Fact 2.6. [EH68, Proof of Theorem 2]
Let ≤ r < ω be a natural number and κ be a cardinal, χ ( Sh symr ( i r − ( κ ))) ≤ κ and χ (cid:16) Sh r ( i r − ( κ ) + ) (cid:17) ≥ κ + . YATIR HALEVI, ITAY KAPLAN, AND SAHARON SHELAH
Proof.
We first show that χ (Sh symr ( i r − ( κ ))) ≤ κ . The proof is by induction on r ≥
2. Suppose r = 2. Let < be the lexicographical order on 2 κ . Let Y = (cid:16) (2 κ ) (cid:17) < be the set of increasing pairs let Y be the complement. By Lemma 2.3(1) it isenough to show that χ ( Sh sym (2 κ ) ↾ Y ) ≤ κ , χ (Sh sym (2 κ ) ↾ Y ) ≤ κ . The proofsfor Y and Y are similar so we prove it just for Y .Given ( x, y ) ∈ Y , let c ( x, y ) = min { i < κ : x ( i ) = y ( i ) } . Suppose that x < y < z ∈ κ are such that c ( x, y ) = c ( y, z ). Then x ∧ y = y ∧ z (where x ∧ y = x ↾ c ( x, y ) ). As x < y it must be that x ( c ( x, y )) = 0 and y ( c ( x, y )) = 1,but then there is no room for z ( c ( x, y )) — contradiction.Now suppose that the claim is true for r. By induction, there is a color-ing d : Sh symr ( i r ( κ )) → κ . Let ψ : Sh symr +1 ( i r ( κ )) → Sh sym (2 κ ) be the fol-lowing homomorphism. Given u = ( u , . . . , u r ) ∈ Sh symr +1 ( i r ( κ )), let ψ ( u ) =( d ( u , . . . , u r − ) , d ( u , . . . , u r )). Note that by choice of d , d ( u , . . . , u r − ) = d ( u , . . . , u r ) . In addition, if u and v are connected in Sh symr +1 ( i r ( κ )), then easily ψ ( u ) and ψ ( v )are distinct (because if not, then ψ ( u ) = ψ ( v ) = ψ ( u ) contradiction) andconnected in Sh sym (2 κ ). Hence we are done by Lemma 2.3(3).As for the second inequality, let c : Sh r ( i r − ( κ ) + ) → κ be a coloring. Thecoloring c induces a coloring on Sh r ( i r − ( κ ) + ), by Erd¨os-Rado, there is a subset U ⊆ i r − ( κ ) + of cardinality κ + such that c ↾ [ U ] r is constant, i.e. every r -tuple ofincreasing elements from U is colored by the same color. As a consequence, therecannot be an edge between any u, v ∈ [ U ] r . Indeed, let u ∈ [ U ] r be any element.Let v ∈ [ U ] r be defined by v ( i ) = u ( i + 1) for 0 ≤ i < r − v ( r −
1) = u (1).They are obviously connected by an edge. (cid:3) Embedding a Shift Graph
The aim of this section is to present some general assumptions on a graph G that will imply that G contains the finite subgraphs of some shift graph.3.1. Reducing Injective Homomorphisms to Homomorphisms.
As a firstresult we prove the following, probably well known, proposition. By Lemma 2.3(3),if there is a homomorphism ϕ : H → G then χ ( H ) ≤ χ ( G ). In particular, if H is a shift graph then there are elementary extensions of G with arbitrary largechromatic numbers. Indeed, one may take elementary extensions of the structure( H, G, ϕ ) and apply Fact 2.6.
Fact 3.1. [ER50, Theorem 1]
Let R be an equivalence relation on ( ω n ) < . Thenthere exists an infinite subset N ⊆ ω and ≤ i < · · · < i m ≤ n − such that for ¯ a, ¯ b ∈ ( N n ) < , ¯ a R ¯ b ⇐⇒ m ^ j =1 a i j = b i j . Proposition 3.2.
Let G = ( V, E ) be a graph and assume there exists an homo-morphism of graphs t : Sh k ( ω ) → G . Then there exists n ≤ k , such that ( † ) G contains all finite subgraphs of Sh n ( ω ) .Consequently, if H is a graph that contains all finite subgraphs of Sh k ( ω ) , forsome k , and t : H → G is a homomorphism of graphs, then there exists some n ≤ k such that G satisfies ( † ).Proof. Assume that t = t ( x , . . . , x k − ). The relation t (¯ a ) = t (¯ b ) for ¯ a, ¯ b ∈ ( ω k ) < ,is an equivalence relation on ( ω k ) < . By Fact 3.1, there exists an infinite subset NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 5 N ⊆ ω and 0 ≤ i < · · · < i m ≤ k − a, ¯ b ∈ ( N k ) < ( †† ) t (¯ a ) = t (¯ b ) ⇐⇒ m ^ j =1 a i j = b i j . Note that m ≥ t (¯ a ) = t (¯ b ) for any ¯ a and ¯ b , but this is impossiblesince there are ¯ a, ¯ b ∈ ( N k ) < that are connected by an edge.Let S = { i , . . . , i m } . There exists a unique set I ⊆ { , . . . , m } and a uniquesequence of natural numbers ¯ n = h n j : j ∈ I i such that S = S j ∈ I [ i j , i j + n j ] andeach interval [ i j , i j + n j ] is maximal with respect to containment.Consider the first-order structure M = (( N, < ) , G = ( V, E ) , t : ( N k ) < → G ).Since ( † ) and ( †† ) are elementary properties, replacing M by an elementary exten-sion, we may assume that ( N, < ) = ( I × Z , < lex ).We define an injective homomorphism Sh n +1 ( ω ) → G , where n = max j ∈ I { n j } .For any f ∈ ( ω n +1 ) < we associate ψ f ∈ V . For that we first define η f ∈ (( I × Z ) k ) < and then set ψ f = t ( η f ). For any j ∈ I and 0 ≤ r ≤ n j we define η f ( i j + r ) = ( j, f ( r )) . For 0 ≤ i ≤ k − i / ∈ S , ( ∗ ) we may set η f ( i ) any way we want provided η f is increasing, which we can since we have copies of Z . Note that this is will notinfluence t ( η f ) by ( †† ).We check that f ψ f is an injective homomorphism. Injectivity: if t ( η f ) = t ( η g )then by ( †† ), η f ( i ) = η g ( i ) for all i ∈ S . In particular for j ∈ I with n j = n and forany 0 ≤ r ≤ n , f ( r ) = g ( r ), as needed.Homomorphism: let f, g ∈ Sh n +1 ( ω ) and assume without loss of generality thatfor every 1 ≤ r ≤ n , f ( r ) = g ( r −
1) and in case n = 0 assume that f (0) < g (0).Using ( ∗ ), we may assume that η f ( i ) = η g ( i −
1) for 1 ≤ i ≤ k −
1. Indeed, considerthe following modification on η g . For every j ∈ I and 0 ≤ r ≤ n j keep η g ( i j + r ) asbefore, and if i j > η g ( i j −
1) = ( j, f (0)) . Note that if n = 0 then η g ( i j −
1) = ( j, f (0)) < ( j, f (1)) = ( j, g (0)) = η g ( i j )and if n = 0 then η g ( i j −
1) = ( j, f (0)) < ( j, g (0)) = η g ( i j ) . Hence η g restricted to S ∪ { i j − j ∈ I, i j > } is increasing. For any other i ∈ I set η g ( i ) any way we want provided η g is increasing.Define η ∈ (( I × Z ) k ) < by η ( i ) = η g ( i − ≤ i ≤ k −
1. If there exists j ∈ I with i j = 0 define η (0) = (0 , f (0)). Notethat then if n = 0 then η (0) = (0 , f (0)) < (0 , f (1)) = (0 , g (0)) = η g (0) = η (1) andif n = 0 then η (0) = (0 , f (0)) < (0 , g (0)) = η g (0) = η (1). Otherwise define η (0) tobe a new element smaller than any element we have encountered in η g . If we showthat η ( i ) = η f ( i ) for all i ∈ S then this would imply that t ( η ) = t ( η f ). Since η and η g are connected by an edge and t is a homomorphism it follows that ψ f = t ( η f )and ψ g = t ( η g ) are connected by an edge, as required.Let j ∈ I and 0 ≤ r ≤ n j . If r ≥ η f ( i j + r ) = ( j, f ( r )) = ( j, g ( r − η g ( i j + r −
1) = η ( i j + r ) . If r = 0 and i j > η f ( i j ) = ( j, f (0)) = η g ( i j −
1) = η ( i j ) . YATIR HALEVI, ITAY KAPLAN, AND SAHARON SHELAH
Finally, if r = 0 and i j = 0 then η f (0) = (0 , f (0)) = η (0) . As for the “consequently” part, consider (
H, t, G ) as a first order structure. Inan elementary extension (
H, t, G ) ≺ ( H , t, G ), H contains Sh n ( ω ) as a subgraph.Restricting t to Sh n ( ω ) and applying the above, G contains all finite subgraphs ofSh n ( ω ) for some n ≤ k . As a result, so does G . (cid:3) Variants of the Shift Graph.
Let A and J be two (possibly linearly ordered)sets. Definition 3.3.
For any ¯ a, ¯ b ∈ A J (respectively, ( A J ) < ), let f ¯ a, ¯ b = { ( i, j ) ∈ J × J : a i = b j } .Since the tuples ¯ a and ¯ b are without repetitions, f ¯ a, ¯ b is a (possibly empty) in-jective partial function. If ¯ a, ¯ b ∈ ( A J ) < then f ¯ a, ¯ b is order-preserving, i.e. for all i < j ∈ Dom( f ), f ( i ) < f ( j ). Definition 3.4.
Let Id = f ⊆ J × J be a partial function. We define a graph E Af and a directed graph D Af on A J : • ¯ a E Af ¯ b ⇐⇒ f ¯ a, ¯ b = f ∨ f ¯ b, ¯ a = f • ¯ a D Af ¯ b ⇐⇒ f ¯ a, ¯ b = f. Similarly for ( A J ) < . We omit A from E Af and D Af when it is clear from the context. Remark . We required f = Id in order to ensure irreflexivity.A homomomorphism between directed graphs is map preserving the directedgraph relation.Since the symmetric closure of the relation D f is exactly E f , any homomorphismof directed graphs ( A J , D f ) → ( A J , D f ) is also a homomorphism of graphs( A J , E f ) → ( A J , E f ), and similarly in the ordered case. Example 3.6.
When J = n and f = { ( i, i −
1) : 1 ≤ i ≤ n − } , ( A n , E f ) is exactlySh symn ( A ) and (( A n ) < , E f ) is exactly Sh n ( A ). Definition 3.7.
Let LSh n ( A ) = (( A n ) < , D f ), where f = { ( i, i −
1) : 1 ≤ i ≤ n − } ,and RSh n ( A ) = (( A n ) < , D f ), where f = { ( i − , i ) : 1 ≤ i ≤ n − } . Lemma 3.8.
Let ( J, < ) be a finite linearly ordered set and Id = f ⊆ J × J anon-empty partial function. Assume that(1) f is order preserving, i.e. for all i < j ∈ Dom( f ) , f ( i ) < f ( j ) ,(2) all orbits in f are increasing , i.e. for all i ∈ Dom( f ) , i < f ( i ) . Let G = (( Q J ) < , D f ) be the directed graph structure defined by f . Then for anylarge enough k ∈ N there exists a homomorphism of directed graphs ϕ : LSh k ( ω ) → (( Q J ) < , D f ) .Proof. Let A be the ordinal ω ω , seen as a substructure of Q . As J is finite, we mayassume that ( J, < ) ⊆ ( Q , < ). Claim.
There is no harm in replacing J by Dom( f ) ∪ Range( f ) and Q by A .Proof. Inductively, for every u ∈ LSh k ( ω ) choose a dense subset Q u ⊆ Q such thatfor every u = v ∈ LSh k ( ω ), Q u ∩ Q v = ∅ and Q u ∩ ω ω = ∅ .Now, let b J = Dom( f ) ∪ Range( f ) and assume we have a homomorphism ϕ :LSh k ( ω ) → (( ω ω ) b J ) < , D f ). For each u ∈ LSh k ( ω ), extending ϕ ( u ) to an in-creasing J -tuple of elements from Q by adding elements from Q u , defines a map ϕ ′ : LSh k ( ω ) → (( Q J ) < , D f ). Since b J = Dom( f ) ∪ Range( f ) and passing from ϕ ( u )to ϕ ′ ( u ) adds only new elements, ϕ ′ is a homomorphism of directed graphs. (cid:3) (claim) NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 7
Let I = Dom( f ) \ Range( f ). We prove by induction on | I | that for any largeenough k there exists a homomorphism g : LSh k ( ω ) → G = (( A J ) < , D f ), where J = Dom( f ) ∪ Range( f ) is nonempty.For any β ∈ I let n β be the maximal natural number n ≥ f n − ( β ) ∈ Dom( f ). Note that Dom( f ) = [ β ∈ I { β, . . . , f n β − ( β ) } and that J = Dom( f ) ∪ Range( f ) = [ β ∈ I { β, . . . , f n β ( β ) } . Let β be the minimal element of I . Note that β is also the minimal element of J . Claim.
There exist J ⊆ e J ⊆ Q and f ⊆ e f ⊆ e J × e J such that • (1) , (2) of the lemma hold for e J and e f , • e f ∩ ( J × J ) = f , • Dom( e f ) \ Range( e f ) = I , • min e J = β and that • ( ⋆ ) letting e n β be the maximal natural number n ≥ such that f n − ( β ) ∈ Dom( e f ) , f e n β ( β ) = max e J .Proof. If ( ⋆ ) holds for f and J , we are done. Otherwise, let i ∈ Range( f ) be minimalsuch that f n β ( β ) < i and let j be such that f ( j ) = i . Either j < f n β ( β ) or f n β ( β ) < j . If the former happens let f ′ = f , so assume it is the latter, i.e. that f n β ( β ) < j < i (so j ∈ Dom( f ) \ Range( f )). Let f ′ = f ∪ { ( f n β ( β ) , y ) } forsome j < y ∈ Q \ J which satisfies y < x for all j < x ∈ J and J ′ = J ∪ { y } . It isstill order preserving and still has increasing orbits. Let n ′ β be as in ( ⋆ ).In either case, we have that j < ( f ′ ) n ′ β ( β ) < i . Since f ′ is order preservingwe may extend it to an automorphism σ of Q and thus i = σ ( j ) < σ (( f ′ ) n ′ β ( β )).By careful adjustments we may assume that σ (( f ′ ) n ′ β ( β )) / ∈ J . Let f ′′ = f ′ ∪{ (( f ′ ) n ′ β ( β ) , σ (( f ′ ) n ′ β ( β ))) } and let J ′′ = J ∪ { σ (( f ′ ) n ′ β ( β )) } . It is still orderpreserving and still has increasing orbits.Note that, |{ i ∈ Range( f ′′ ) : i < ( f ′′ ) n ′′ β ( β ) }| < |{ i ∈ Range( f ) : i < f n β ( β ) }| , where n ′′ β is defined as is ( ⋆ ).Continue doing this until Range( f ) is exhausted. Let e f be the end function andlet e J = J ∪ Dom( e f ) ∪ Range( e f ). (cid:3) (claim)As a consequence of the claim we may assume that ( ⋆ ) holds for f and J . Indeed,assume we found a homomorphism ϕ : LSh k ( ω ) → (( A e J ) < , D e f ), for some k . It isroutine to check that composing ϕ with the map(( A e J ) < , D e f ) → (( A J ) < , D f )induced by the projection map π : ( A e J ) < → ( A J ) < is indeed a homomorphism(this uses the second bullet in the claim above).Let J ′ = J \ { β , . . . , f n β ( β ) } . If J ′ = ∅ let g k be the empty function for all k ∈ N . Otherwise, by induction there exists l ∈ N such that for all k ≥ l there isa homomorphism g k : LSh k ( ω ) → (( A J ′ ) < , D f ∩ ( J ′ × J ′ ) ). Let k > max { n β + 1 : β ∈ I } ∪ { l } and set some order isomorphism φ : ω × ( A ∪ {− } ) → A , where − A (recall that A = ω ω ). YATIR HALEVI, ITAY KAPLAN, AND SAHARON SHELAH
We construct a homomorphism mapping µ ∈ LSh k ( ω ) to ψ µ ∈ G . Let µ ∈ LSh k ( ω ). For any 0 ≤ h ≤ n β we define ψ µ ( f h ( β )) = φ ( µ ( h ) , − . For any β ∈ I , with β = β , and 0 ≤ h ≤ n β we define ψ µ ( f h ( β )) = φ ( µ (˜ h ) , g k ( µ )( f h ( β ))) , for 0 ≤ ˜ h ≤ n β maximal satisfying f h ( β ) > f e h ( β ), which exists by minimlaity of β .We check that ψ µ is increasing and that µ ψ µ is a homomorphism.To show that ψ µ is increasing, suppose f h ( β ) < f h ( β ) ∈ J and go over thedifferent possibilities. Note that we use − β = β , β = β and e h = h .We show that µ ψ µ is a homomorphism. Suppose that µ, ν ∈ LSh k ( ω ) aresuch that ν ( n + 1) = µ ( n ) for all 0 ≤ n < k −
1. We need to check that f ( i ) = j ifand only if ψ µ ( i ) = ψ ν ( j ).Assume that f ( i ) = j (so i ∈ Dom( f )). Suppose that i = f h ( β ) for some0 ≤ h < n β , so j = f h +1 ( β ). Then ψ µ ( i ) = φ ( µ ( h ) , −
1) = φ ( ν ( h + 1) , −
1) = ψ ν ( f h +1 ( β )) . Now suppose that i = f h ( β ) for some β = β and 0 ≤ h < n β , so j = f h +1 ( β ).Let e h be maximal such that f e h ( β ) < f h ( β ). Note that by the claim, e h < n β . Itfollows that f h +1 ( β ) is defined and f e h +1 ( β ) < f h +1 ( β ). On the other hand, it cannot be that f e h +2 ( β ) < f h +1 ( β ) (again, f e h +2 ( β ) is defined by the claim) for thenwe would have f e h +1 ( β ) < f h ( β ), contradicting the maximality of e h . It followsthat e h + 1 = ] h + 1. Since g k is a homomorphism, ψ µ ( i ) = φ ( µ ( e h ) , g k ( µ )( f h ( β ))) = φ ( ν ( e h + 1) , g k ( ν )( f h +1 ( β )) = ψ ν ( j ) . Now assume that ψ µ ( i ) = ψ ν ( j ). If i = f h ( β ) for some 0 ≤ h ≤ n β then j = f h ′ ( β ) for some h ′ (since ψ µ ( j ) has the form φ ( − , − µ ( h ) = ν ( h ′ ).By the choice of the k , h + 1 < k and consequently µ ( h ) = ν ( h + 1) = ν ( h ′ ) so h ′ = h + 1 and f ( i ) = j .Suppose i = f h ( β ) for some β = β and 0 ≤ h ≤ n β . As this is encoded by φ , by the assumption necessarily j = f h ′ ( β ′ ) for some β ′ = β and 0 ≤ h ′ ≤ n β ′ .Let e h be maximal such that f e h ( β ) < i and e h ′ maximal such that f e h ′ ( β ) < j . So ψ µ ( i ) = φ ( µ ( e h ) , g k ( µ )( i )) and ψ ν ( j ) = φ ( ν ( e h ′ ) , g k ( ν )( j )). It follows that g k ( ν )( j ) = g k ( µ )( i ) and we are done by the choice of g k since i, j ∈ J ′ . (cid:3) Before continuing to the main proposition, as auxiliary results, we calculate thechromatic number of some (well known) graphs.
Example 3.9 (Symmetric Cyclic Graph) . Let r > symr ( A ) be the graph on A r with an edge between ( a , . . . , a r − ) and ( b , . . . , b r − )if a = b , . . . , a r − = b r − , a r − = b (or vice-versa). We thus have a graph homo-morphism Cyc symr ( A ) → Sh symr ( A ). Lemma 3.10.
For every natural number r > and any set A , χ ( Cyc symr ( A )) = ( r is even r is odd . Proof.
The graphs Cyc symr ( A ) partitions into connected components, each of thema cycle graph on r vertices. It is well known and easy to see that you need 2 colorsto color even cycle graphs and 3 colors to color odd cycle graphs. (cid:3) NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 9
The next two examples are somewhat similar and they both have very smallchromatic number. We define them and prove that their chromatic number is 2.
Example 3.11 (Denumerable Tuples Symmetric Shift Graph) . Let A be an infiniteset. The denumerable tuples symmetric shift graph Sh symω ( A ) is defined similarlyas the symmetric shift graph but with vertices A ω . There is an edge between twovertices f and g if f ( n ) = g ( n + 1) for all n < ω (or vice-versa). Example 3.12 (Glued Increasing Symmetric Shift Graphs) . Let ¯ n = h n i : i < ω i be a strictly increasing sequence of natural numbers. We define the graph Sh sym ¯ n,u ( A ),for an infinite set A . The vertices are injective functions ` i<ω [0 , n i ] → A . Thusevery vertex can be written as f = ` i<ω f i . We will say that there is an edgebetween two vertices f and g if f i ( m ) = g i ( m + 1)for every 0 ≤ m < n i , i < ω or g i ( m ) = f i ( m + 1)for every 0 ≤ m < n i , i < ω . Lemma 3.13.
Let X be a set. Suppose ( n, x ) n + x is a free action of Z on X . We define a graph relation E on X by setting that x E y if either x = y or y = x . Then χ ( X, E ) = 2 .Proof.
Since the action is free, for any x, y ∈ X in the same Z -orbit, D x,y = z forany z ∈ Z satisfying z + x = y is well defined.Let { x r : r ∈ X/ Z } be a set of representatives of the different orbits of the Z action on X . We define a coloring c : ( X, E ) → { , } in the following way: Forany vertex x let c ( x ) = parity( D x,x r ), where x r is the representative of the Z -orbitof x .We need to show that it is a legal coloring. Assume that x and y are connectedby an edge. This means, without loss of generality, that x = 1 + y . But nowby definition we cannot have c ( x ) = c ( y ). Indeed, since x and y are in the same Z -orbit, there are some n and m such that x = n + x r and y = m + x r , for r = Z + x . Consequently, x = ( n − m ) + y , and by freeness n − m = 1. As a result, c ( x ) ≡ c ( y ) + 1(mod 2). (cid:3) Lemma 3.14.
For any infinite set A and a strictly increasing sequence of naturalnumbers ¯ n , χ ( Sh symω ( A )) = χ (cid:0) Sh sym ¯ n,u ( A ) (cid:1) = 2 .Proof. The proof for these two graphs are the same, albeit the definitions areslightly different. We prove for Sh symω ( A ) and present the appropriate definitionsfor Sh sym ¯ n,u ( A ) at the end.Let X ⊆ A ω be the set of all functions f which are eventually injective, i.e. thereexists an n such that f ↾ [ n, ∞ ) is injective.Fix some element e ∈ A . The integers Z acts on X by translation: if z ∈ Z and f ∈ X then we define ( z + f )( m ) = ( f ( m − z ) 0 ≤ m − ze otherwiseWe define an equivalence relation R on X : f R g ⇐⇒ ∃ n ( f ↾ [ n, ∞ ) = g ↾ [ n, ∞ )) . Note that if f R g and z ∈ Z then z + f R z + g , so the Z -action induces an actionon X/R . We note that if z + [ f ] = [ f ] for f ∈ X (and [ f ] being the class of f in X/R ) then z = 0 by eventual injectivity of f . Or in other words, the Z -action on X/R is free.Since if f, g ∈ Sh symω ( A ) are connected by an edge then either [ f ] = 1 + [ g ] or[ g ] = 1 + [ f ], by Lemma 3.13 and Lemma 2.3(3), χ (Sh symω ( A )) = 2.For Sh sym ¯ n,u ( A ) we define:Let X be the set of all functions f : ` i<ω → [0 , n i ] satisfying the property thatthere exists an n such that for all i < ω , f i ↾ [ n, n i − n ] is injective.For every z ∈ Z and f ∈ X we define for i < ω ( z + f ) i ( m ) = ( f i ( m − z ) 0 ≤ m − z ≤ n i e otherwiseWe define an equivalence relation R on X : f R g ⇐⇒ ∃ n ∀ i < ω ( f i ↾ [ n, n i − n ] = g i ↾ [ n, n i − n ]) . (cid:3) On the other hand if the glued shift graphs are bounded the picture is different.
Example 3.15 (A Sequence of Bounded Shift Graphs) . Let n be a natural number, I a set and ¯ n = h n i : i ∈ I i a sequence of natural numbers satisfying 0 < n i ≤ n for all i ∈ I . We define Sh ¯ n,b ( A ) for an infinite linearly ordered set ( A, < ). Thevertices are sequences of functions f = ( f i ) i ∈ I , where each f i : [0 , n i ] → A is orderpreserving. We will say that there is an edge between two vertices f and g if f i ( m ) = g i ( m + 1) for every 0 ≤ m < n i , i ∈ I (or vice-versa). Lemma 3.16.
Let ( A, < ) be an infinite linearly ordered set, ¯ n = h n i : i ∈ I i auniformly bounded sequence of natural numbers with n i ≥ and let n = max i ∈ I { n i } .Then there exists an injective homomorphism Sh n +1 ( A ) → Sh ¯ n,b ( A ) .Proof. For any tuple u ∈ ( A n +1 ) < we define a vertex f u ∈ Sh ¯ n,b ( A ). For every 0 ≤ h ≤ n i , i ∈ I , we set ( f u ) i ( h ) = u ( h ). Set f = ( f i ) i ∈ I . Note that if 0 ≤ h < h ′ ≤ n i then u ( h ) < u ( h ′ ) so ( f u ) i ( h ) < ( f u ) i ( h ′ ). By the choice of n , u f u is injectiveas well.We show that u f u is a homomorphism. Assume that, without loss of gen-erality, u ( h ) = v ( h + 1) for every 0 ≤ h < n . For every i ∈ I and for every0 ≤ h < n i ( f u ) i ( h ) = u ( h ) = v ( h + 1) = ( f v ) i ( h + 1) . As needed. (cid:3)
The following propositions will be the backbone behind the main results.
Proposition 3.17.
Let A be an infinite set, λ a cardinal with λ ≤ | A | and G =( A λ , E ) a graph on A λ . If κ is an infinite regular cardinal satisfying(1) i ( λ ) < κ ,(2) χ ( G ) ≥ κ and(3) for all ¯ a, ¯ b, ¯ c, ¯ d ∈ A λ if ¯ a E ¯ b and f ¯ a, ¯ b = f ¯ c, ¯ d then ¯ c E ¯ d then there exists an n ∈ N and an injective homomorphsim from Sh n ( ω ) to G .Proof. Let F = { f ¯ a, ¯ b : ¯ a E ¯ b } be the collection of all functions arising as f ¯ a, ¯ b forsome ¯ a and ¯ b sharing an edge. If we set E f = { (¯ a, ¯ b ) : f = f ¯ a, ¯ b ∨ f = f ¯ b, ¯ a } then,since by assumption (3), E = S f ∈ F E f , then by Lemma 2.3(2) κ ≤ Y f ∈ F χ ( V, E f ) . NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 11
Since | F | ≤ λ we may assume there exists f ∈ F with χ ( V, E f ) > λ ≥ ℵ .Replace G by ( V, E f ). Note that although now we only have that χ ( V, E ) ≥ ℵ , wegained that ¯ a and ¯ b are connected by an edge if and only if f ¯ a, ¯ b = f or f ¯ b, ¯ a = f .For any β ∈ Dom( f ) ⊆ λ , we distinguish between four possibilities:(1) “ β is a fixed point”: f ( β ) = β ;(2) “ β generates a finite cycle”: there exists a natural number 1 < n ∈ N suchthat f n ( β ) = β and f n − ( β ) = β ;(3) “ β generates a finite shift”: there exists a natural number 0 < n ∈ N suchthat f n ( β ) / ∈ Dom( f );(4) “ β generates an infinite shift”: the set { f n ( β ) : n < ω } is infinite.We first note the following observations, which will allow us to cross out someof the possibilities: • No β ∈ Dom( f ) generates a finite cycle. Assume there exists β ∈ Dom( f )and 1 < n < ω such that f n ( β ) = β and f n − ( β ) = β . Define a ho-momorphism G → Cyc symn ( A ) which maps ¯ a to ( a β , a f ( β ) , . . . a f n − ( β ) ). Itis a homomorphism because if there is an edge between ¯ a and ¯ b then bydefinition of f , a β = b f ( β ) , . . . , a f n − ( β ) = b β . By Lemma 3.10 and Lemma2.3(3), χ ( G ) ≤ χ (Cyc symn ( A )) ≤
3, contradiction. • No β ∈ Dom( f ) generates an infinite shift. Assume there exists β < λ with { f n ( β ) : n < ω } infinite. Define a homomorphism G → Sh symω ( A ) whichmaps ¯ a to n a f n ( β ) . By definition this is a homomorphism of graphs. ByLemma 3.14 and Lemma 2.3(3), χ ( G ) ≤ χ (Sh symω ( A )) = 2, contradiction.Let I = Dom( f ) \ Range( f ). For any β ∈ I let n β be the maximal naturalnumber n ≥ f n − ( β ) ∈ Dom( f ). Note thatDom( f ) = [ β ∈ I { β, . . . , f n β − ( β ) } ∪ { β < λ : f ( β ) = β } and that Dom( f ) ∪ Range( f ) = [ β ∈ I { β, . . . , f n β ( β ) } ∪ { β < λ : f ( β ) = β } . Claim.
There exists a uniform bound on { n β : β ∈ I } .Proof. Otherwise, assume there are h β i : i < ω i such that the sequence ¯ n := h n β i : i < ω i is strictly increasing. We define a homomorphism from G to Sh sym ¯ n,u ( A )similarly as before. Consequently, χ ( G ) ≤ (cid:3) (claim)We are thus left with two cases:Case 1: I = ∅ . Thus f is the identity on Dom( f ). If Dom( f ) = λ then G isan anticlique and can thus can be colored by only one color, contradiction. HenceDom( f ) ( λ . Let ¯ a ∈ G be any sequence. Consider the induced subgraph G ⊆ G with vertices: { ¯ b ∈ G : ( ∀ i ∈ Dom( f )) ( b i = a i ) } . By the definition of the edge relation G is a complete graph of size | A | . In particularwe may embed the complete graph on ω as a subgraph.Case 2: I = ∅ . Let n = max β ∈ I { n β } and let φ : λ × ω × (Sh ¯ n,b ( ω ) ∪ { , } ) → A be an injective function, which exists since ( ℵ ) λ + ℵ + λ ≤ | A | .We define an injective homomorphism from Sh ¯ n,b ( ω ) into G where ¯ n = h n β : β ∈ I i . For every function µ = ( µ β ) β ∈ I , where µ β : [0 , n β ] → ω is order preserving,we associate an injective function ψ µ : λ → A as follows. For every β ∈ I and h ∈ [0 , n β ] we define ψ µ ( f h ( β )) = φ ( β, µ β ( h ) , , note that this is well defined. For every α S β ∈ I { β, . . . , f n β ( β ) } such that f ( α ) = α we define ψ µ ( α ) = φ ( α, , ψ µ ( α ) = φ ( α, , µ ) . We claim that the map µ ψ µ is an injective homomorphism.Injectivity: Let µ, ν ∈ Sh ¯ n,b ( ω ) with ψ µ = ψ ν . Let β ∈ I and h ∈ [0 , n β ]. Since ψ µ ( f h ( β )) = ψ ν ( f h ( β )) and φ is injective, µ β ( h ) = ν β ( h ).Homomorphism: Assume that µ and ν are connected by an edge, i.e. withoutloss of generality for every β ∈ I and h ∈ [0 , n β ), µ β ( h ) = ν β ( h + 1). We need toshow that for every i, j < λ , ψ µ ( i ) = ψ ν ( j ) if and only f ( i ) = j .Assume that f ( i ) = j . In particular, i ∈ Dom( f ). If i = β = f ( β ) = j then ψ µ ( β ) = φ ( β, ,
1) = ψ ν ( β ). If i ∈ Dom( f ) \ Range( f ) then i = f h ( β ) for some β ∈ I and h ∈ [0 , n β ). Thus ψ µ ( i ) = ψ µ ( f h ( β )) = φ ( β, µ β ( h ) ,
0) = φ ( β, ν β ( h + 1) ,
0) = ψ ν ( f h +1 ( β )) = ψ ν ( j ) . Assume that ψ µ ( i ) = ψ ν ( j ) = e . Since µ = ν , by the injectivity of φ we haveonly two possibilities: either e = φ ( · · · ,
1) or e = φ ( · · · , f ( i ) = i , f ( j ) = j and i = j .Otherwise, i = f h ( β ) and j = f h ′ ( β ′ ) for some β, β ′ ∈ I , h ∈ [0 , n β ] and h ′ ∈ [0 , n ′ β ]. Also, since φ ( β, µ β ( h ) ,
0) = φ ( β ′ , ν β ′ ( h ′ ) , ,β = β ′ and µ β ( h ) = ν β ( h ′ ). If h ∈ [0 , n β ) then µ β ( h ) = ν β ( h + 1) since µ and ν areconnected by an edge, so since ν β is injective h ′ = h + 1. So f ( i ) = j . Otherwise, h = n β . If h ′ > ν β ( h ′ ) = µ β ( h ′ −
1) = µ β ( h ) we get a contradictionto the injectivity of µ β . Consequently it must be that h ′ = 0. This contradicts thefact that ν β is order preserving and n β ≥ n +1 ( ω ) into Sh ¯ n,b ( ω ), and thus into G as well. (cid:3) Proposition 3.18.
Let ( A, < ) be an infinite linearly ordered set, m < ω and G = (( A m ) < , E ) a graph on ( A m ) < . Assume χ ( G ) ≥ ℵ and that for all ¯ a, ¯ b, ¯ c, ¯ d ∈ ( A m ) < if ¯ a E ¯ b and f ¯ a, ¯ b = f ¯ c, ¯ d then ¯ c E ¯ d . Then there exists n < ω such that G contains all finite subgraphs of Sh n ( ω ) .Proof. As was done in the proof of Proposition 3.17, letting F = { f ¯ a, ¯ b : ¯ a E ¯ b } ,since | F | < ℵ , we may assume that E = E f for some f ∈ F (see Definition 3.4).Since the tuples are increasing, f is necessarily an order preserving function( i < j ∈ Dom( f ) = ⇒ f ( i ) < f ( j )). Thus, as m is finite, for any β ∈ Dom( f ) ⊆ m with f ( β ) = β , “ β generates a finite shift” (in the context of Proposition 3.17), i.e.there exists a natural number 0 < n ∈ N such that f n ( β ) / ∈ Dom( f ).Let I = Dom( f ) \ Range( f ). For any β ∈ I let n β be the maximal naturalnumber n ≥ f n − ( β ) ∈ Dom( f ). Note thatDom( f ) = [ β ∈ I { β, . . . , f n β − ( β ) } ∪ { β < m : f ( β ) = β } and thatDom( f ) ∪ Range( f ) = [ β ∈ I { β, . . . , f n β ( β ) } ∪ { β < m : f ( β ) = β } . As in the proof of Proposition 3.17(Case 1), we may assume that I = ∅ . Say that β ∈ I is increasing if β < f ( β ) and decreasing otherwise (equivalently, f ( β ) < β ). NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 13
Also, as f is order preserving and the tuples are increasing, we may find a partition m = J ∪ · · · ∪ J N satisfying that • each of the J i are convex and J < · · · < J N ; • if β ∈ J i ∩ Dom( f ) then f ( β ) ∈ J i ; • if β ∈ I ∩ J i is increasing then every β ′ ∈ I ∩ J i is increasing; • if β ∈ I ∩ J i is decreasing then every β ′ ∈ I ∩ J i is decreasing and • if for β ∈ J i , f ( β ) = β then for every β ′ ∈ (Dom( f ) ∪ Range( f )) ∩ J i , β = β ′ .For every 1 ≤ i ≤ N , set f i = f ∩ ( J i × J i ).For any 1 ≤ i ≤ N if J i is of increasing type, by applying Lemma 3.8 there is ahomomorphism g i,k : LSh k ( ω ) → (( Q J i ) < , D f i ) for any large enough k .For any 1 ≤ i ≤ N if J i is of decreasing type, by applying Lemma 3.8 to( Q , < ∗ ) (there reverse order on Q ) and ( J i , < ∗ ) (the reverse order on J i ) there is ahomomorphism g ∗ i,k : LSh k ( ω ) → (( Q ( J i ,< ∗ ) ) < ∗ , D f i ) for any large enough k . Sincethe identity function is an isomorphism of directed graphs(( Q ( J i ,< ∗ ) ) < ∗ , D f i ) ∼ = (( Q J i ) < , D f i ) , we may compose an get a homomorphism g i,k : LSh k ( ω ) → (( Q J i ) < , D f i ).Let k be large enough so that g i,k are defined for all i and set g i = g i,k .For any 1 ≤ i ≤ N , if J i is of constant type fix some embedding g i : ( J i , < ) → ( Q , < ).Let ( A, < ) ≺ ( A , < ) be a sufficiently saturated extension with ( A , < ) containing( Q, < ) = ( N × Q × (LSh k ( ω ) ∪ { } ) , < ) as a substructure, where we may choseany linear order on LSh k ( ω ) ∪ { } . Since the inclusion ( Q, < ) ⊆ ( A , < ) induces aninjective homomorphism (( Q m ) < , D f ) → (( A m ) < , D f ) , we may assume that A = Q . We will construct a homomorphism LSh k ( ω ) → (( A m ) < , D f ).Let µ ∈ LSh k ( ω ). We define ψ µ ∈ ( A m ) < as follows. If α ∈ J i , with J i increasingor decreasing then ψ µ ( α ) = ( i, g i ( µ )( α ) , . If α ∈ J i , with J i of constant type, and f ( α ) = α then ψ µ ( α ) = ( i, g i ( α ) , . If α ∈ J i , with J i of constant type, and f ( α ) = α then ψ µ ( α ) = ( i, g i ( α ) , µ ) . Since J < · · · < J N then by definition, ψ µ is increasing. We claim that µ ψ µ isa homomorphism.Assume that µ, ν ∈ LSh k ( ω ) are such that µ ( h ) = ν ( h + 1) for 0 ≤ h < k − f ( α ) = β if and only if ψ µ ( α ) = ψ ν ( β ).If f ( α ) = β then α, β ∈ J i for some 1 ≤ i ≤ N . If J i is not of constant type thensince g i is a homomorphism, g i ( µ )( α ) = g i ( ν )( β ), so ψ µ ( α ) = ( i, g i ( µ )( α ) ,
0) = ( i, g i ( ν )( β ) ,
0) = ψ ν ( β ) . If J i is of constant type then β = f ( α ) = α and ψ µ ( α ) = ( i, g i ( α ) ,
0) = ( i, g i ( β ) ,
0) = ψ ν ( β ) . Now assume that ψ µ ( α ) = ψ ν ( β ). By definition, α, β ∈ J i for some 1 ≤ i ≤ N .If J i is of constant type then since µ = ν by the definition of ψ , f ( α ) = α , f ( β ) = β and g i ( α ) = g i ( β ). Consequently, α = β and as a result α = β = f ( α ). If J i is not of constant type then g i ( µ )( α ) = g i ( ν )( β ). By the fact that g i is ahomomorphism, f ( α ) = β .Finally, by Proposition 3.2, G contains all finite subgraphs of Sh k ( ω ). (cid:3) Superstable and ω -stable Graphs We use the main result of the previous section in order to prove the strong fromof Taylor’s conjecture for ω -graphs and a suitable variant for superstable graphs.The following result is somewhat reminiscent (in flavor) of [ER50, TheoremIII]. It is a local version of the the well known fact that, in stable theories, everyindiscernible sequence is an indiscernible set [TZ12, Lemma 9.1.1] (it is possiblyknown, but could not find a reference).Generalizing the notation from Definition 3.3, for two tuples, possibly of differentlength, ¯ a and ¯ b , we denote f ¯ a, ¯ b = { ( i, j ) : a i = b j } .Recall that for a set of formulas ∆, a ∆ -indiscernible sequence is a sequence ofelements that are indiscernible only with respect to formulas from ∆. For a formula ϕ ( x , . . . , x n − ) let ∆ ϕ := { ϕ ( x π (0) , . . . , x π ( n − ) : π is a function fron n to n } . Proposition 4.1.
Let T be a complete theory and ϕ ( x, y ) a partitioned stable for-mula, with x and y possibly of different lengths. Let I be a ∆ ϕ -indiscernible sequenceindexed by an infinite linearly ordered set ( Q, < ) .If there exist ¯ a ∈ I | x | and ¯ b ∈ I | y | disjoint increasing tuples such that ϕ (¯ a, ¯ b ) holds then for every disjoint increasing tuples ¯ c ∈ I | x | and ¯ d ∈ I | y | , ϕ (¯ c, ¯ d ) holds.Moreover, if ¯ a, ¯ c ∈ I | x | and ¯ b, ¯ d ∈ I | y | are increasing tuples then ( ⋆ ) ϕ (¯ a, ¯ b ) ∧ f ¯ a, ¯ b = f ¯ c, ¯ d = ⇒ ϕ (¯ c, ¯ d ) . Proof.
Let I ′ be an indiscernible sequence with the same EM-type as I . Suppose( ⋆ ) is not true as witnessed by ¯ a, ¯ b, ¯ c, ¯ d , then let ¯ a ′ , ¯ b ′ , ¯ c ′ , ¯ d ′ in I ′ be such that ¯ a ′ ¯ b ′ as the same order type as ¯ a ¯ b , and ¯ c ′ ¯ d ′ has the same order type as ¯ c ¯ d . It followsthat ( ⋆ ) is not true for ¯ a ′ , ¯ b ′ , ¯ c ′ , ¯ d ′ . We may thus assume that I is an indiscerniblesequence. Similarly, we may assume that ( Q, < ) is a dense linear order with noendpoints. Also, we endow I with the order induced by Q , i.e. we write a i < a j but mean i < j ∈ Q .Let ¯ a, ¯ b, ¯ c and ¯ d be as in the statement. By applying an automorphism we mayassume that ¯ c = ¯ a . Let X = { ¯ c ∈ I \ ¯ a : ϕ (¯ a, ¯ c ) } . Note ¯ b ∈ X .By stability the ϕ -type tp ϕ (¯ a/I \ ¯ a ) is definable, i.e. there is some formula ψ ( y, ¯ e )with ¯ e ∈ I \ ¯ a such that ¯ c ∈ X ⇐⇒ ¯ c | = ψ ( y, ¯ e ) . Let ¯ h ∈ I \ ¯ a ¯ e have the same order type as ¯ e over ¯ a , which exists by density. Let σ be an automorphism of ( I, < ) which fixes ¯ a and maps ¯ e to ¯ h . By indiscernibilityit follows that σ ( X ) = X .We claim that X is definable over both ¯ e and ¯ h in the structure ( I \ ¯ a, < ). Indeedif ¯ c , ¯ c have the same order type over ¯ e then ¯ c ∈ X ⇐⇒ ¯ c ∈ X . Since thereonly finitely many order types over ¯ e , this shows the claim for ¯ e . As σ ( X ) = X , wehave it also for ¯ h .As DLO eliminates imaginaries [TZ12, Exercise 8.4.3], X has a code p X q ∈ dcl(¯ e ) ∩ dcl(¯ h ). As dcl is trivial in DLO and ¯ e and ¯ h are disjoint, X is definableover ∅ . Consequently, either X is empty or equals to I \ ¯ a . Since X is not empty, X = I \ ¯ a . This proves the first part.Now for the moreover part. Assume that ϕ (¯ a, ¯ b ) holds and f ¯ a, ¯ b = f ¯ c, ¯ d . Byapplying an automorphism, we may assume that ¯ c = ¯ a . We prove it by inductionon | Dom( f ) | , where f = f ¯ a, ¯ b = f ¯ a, ¯ d , the case | Dom( f ) | = 0 being the first part ofthe proposition. NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 15
Assume this is true for functions whose domains have cardinality less than k andthat | Dom( f ) | = k . Let i be the maximal element of Dom( f ).Note that I f ( i ) ) (because d f ( i ) = a i = b f ( i ) ). Now note that I >a i is also indiscernible over I ≤ a i and we consider theformula ψ ( u, v ) = ϕ ( a ≤ i u, d ≤ f ( i ) v ) (recall that d f ( i ) = a i ). Applying the base ofthe induction hypothesis, we conclude that ϕ (¯ a, ¯ d ), as required. (cid:3) Definition 4.2.
Let L be a first order language and T a complete L -theory withinfinite models and let ∆ be a set of formulas. An EM ∆ -Model of T is a modelwhich is generated by a ∆-indiscernible sequence, i.e. a model M | = T with a ∆-indiscernible sequence I such that for every b ∈ M there exist a term t ( x , . . . , x n − )and elements a < · · · < a n − ∈ I with b = t ( a , . . . , a n − ). If ∆ is the set of allformulas we omit ∆ from the notation.For a binary relation E , let ∆( E ) be the collection of formulas of the form E ( t ( x ) , t ( y )), where t is a term. Theorem 4.3.
Let L = { E, . . . } be a first order language with E a binary relation.Let T an L -theory specifying that E is a symmetric and irreflexive stable relation.Let κ > | T | + ℵ be an infinite regular cardinal. Let G = ( V ; E, . . . ) | = T be anEM ∆( E ) -model. If χ ( V, E ) ≥ κ then there exists a natural number n such that G contains all finite subgraphs of Sh n ( ω ) .Proof. Let (
A, < ) be a linearly ordered set, I = h r i : i ∈ A i an indiscerniblesequence and { t α } α< | T | a set of terms satisfying that V = S α< | T | t α ( I ), where t α ( I ) is the image of the map I V given by substituting increasing tuples in t α .By Lemma 2.3(1), κ ≤ χ ( G ) ≤ P α< | T | χ ( t α ( I ) , E ↾ t α ( I )). Since κ is a regularcardinal, there exists an α such that χ ( t α ( I ) , E ↾ t α ( I )) ≥ κ . We may thus assumethat V = t ( I ) for some term t = t (¯ x ) = t ( x , . . . , x n − ).The map t : ( I n ) < → t ( I ) induces a graph on ( I n ) < by specifying that ¯ a E ¯ b ifand only if t (¯ a ) E t (¯ b ). By Lemma 2.3(4), χ (( I n ) < ) ≥ κ as well.Since the edge relation E ( v, u ) is stable, so is E ( t (¯ x ) , t (¯ y )). As a result, themoreover part of Proposition 4.1 allows us to apply Proposition 3.18. Hence ( I n ) < contains all finite subgraphs of Sh k ( ω ) for some k . We may now conclude byapplying the consequently part of Proposition 3.2. (cid:3) Corollary 4.4.
Let G = ( V, E ) be a graph. If • G is superstable and χ ( G ) > ℵ or • G is ω -stable and χ ( G ) > ℵ then G contains all finite subgraphs of Sh n ( ω ) for some n ∈ N .Proof. Suppose G is superstable and χ ( G ) > ℵ . By [She, Claim 16.2(2B.c)] or[Mar01, page 345], [Mar99, Theorem 3.B] there exists { E } ⊆ L of cardinality 2 ℵ and an L -saturated EM-model G such that Th( G ) ↾ { E } = Th( G ). Since G issaturated, we may embed G as an induced subgraph of G . Since χ ( G ) > ℵ , byTheorem 4.3 all finite subgraphs of Sh n ( ω ) are contained in G for some n ∈ N . Theresult now follows since G ≺ G , as graphs.For ω -stable graphs we may use [Mar01, Theorem C] to find an L -saturatedEM-model in a countable language. (cid:3) Stationary Stable Graphs
The crucial part of the proof of Theorem 4.3 was the existence of a saturatedEM-model. It is a natural question to ask whether the technique from the previoussection can be generalized to any stable graph, i.e. is the following true:There exists a cardinal κ such that for every stable graph with χ ( G ) ≥ κ there exists a saturated EM-model G , in an expansion { E } ⊆ L with |L| < κ , such that G ≺ G ↾ { E } .However, Mariou has shown in [Mar99, Theorem 3.A] that if a stable theory T has a κ + -saturated EM-model in an expansion L with |L| ≤ κ then T is superstable.As a result, such a result would imply superstability. For general stable graphs adifferent approach is needed.A connected notion to that of EM-models is the that of representations of struc-tures from [CS16]. We will need a variation on the theme. Definition 5.1 (The Free Algebra) . Suppose A is a pure set. Let M µ,κ ( A ) bethe (non first order) structure whose vocabulary is L µ,κ = { F α,β : α < µ, β < κ } ,where each F α,β is a β -ary function symbol for all α < µ (note that we allow infinitearity). The universe of M µ,κ ( A ) is S γ ∈ Ord M µ,κ,γ ( A ) . Where • M ( A ) = A , • for limit γ , M β ( A ) = S γ ′ <γ M γ ′ ( A ), • and for successor M γ +1 = M γ ∪ { F α,β (¯ b ) : ¯ b ∈ ( M γ ) β , α < µ, β < κ } . We treat F α,β (¯ b ) as a new formal object.For a cardinal κ , let reg( κ ) be κ + if κ is singular and κ otherwise. Fact 5.2. [CS16, Remark 2.3]
Let A and M µ,κ ( A ) be as before. M µ,κ ( A ) is a setwhose cardinality is at most ( | A | + µ ) < reg( κ ) (though defined as a class).Remark . Fixing a set of variables X = { x i : i < reg( κ ) } , the set of terms in L µ,κ in X can be identified with M µ,κ ( X ). It follows from Fact 5.2 that theirnumber is bounded by (reg( κ ) + µ ) < reg( κ ) .For any permutation π of A we denote by b π the induced automorphism of M µ,κ ( A ). Definition 5.4.
Let M be a structure. A homogeneous representation of M in M µ,κ ( A ) is a function Φ : M → M µ,κ ( A ) satisfying(1) For every term t (¯ x ), where ¯ x is tuple of length β < κ containing the vari-ables of t , if t (¯ a ) ∈ Im(Φ) for some ¯ a ∈ A β then t (¯ b ) ∈ Im(Φ) for all¯ b ∈ A β ;(2) For any two finite sequences ¯ a, ¯ b ∈ M n , if there exists an permutation π of A such that b π (Φ( a i )) = Φ( b i ), for all i < n , thentp M (¯ a ) = tp M (¯ b ) . We say that Φ is a skeletal homogeneous representation if it is an injective partialfunction satisfying (1) and (2) on its domain and that dcl(Dom(Φ)) = M . Remark . Representations were originally defined in [CS16, Definition 2.1] andthe definition was that of a function Φ : M → M µ,κ ( A ) satisfying thatqftp(Φ(¯ a )) = qftp(Φ(¯ b )) = ⇒ tp M (¯ a ) = tp M (¯ b ) . Since every permutation of A lifts to an automorphism of the free algebra, theantecedent in condition (2) implies that Φ(¯ a ) and Φ(¯ b ) have the same quantifier-free type. As a result, every representation satisfies condition (2) of a homogeneousrepresentation. NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 17
Proposition 5.6.
Let M be a structure and assume there exists a skeletal homoge-neous representation Φ : Dom(Φ) → M µ,κ ( A ) of M , where A is a pure set, κ and µ are infinite, and that(1) (reg( κ ) + µ ) < reg( κ ) < κ ,(2) i ( λ ) < κ for all λ < reg( κ ) ,(3) < reg( κ ) ≤ | A | .For every graph G = ( V, E ) that is ∅ -interpretable in M with χ ( G ) ≥ κ , where κ is a regular cardinal, there exists an n ∈ N such that G contains all finite subgraphsof Sh n ( ω ) .Proof. Since G is interpretable in M there exist r ∈ N , a definable subset V ⊆ M r and an interpretation g : V → V (see [Hod93, Section 5.3]). By definition, g is surjective and G = ( V , g − ( E )) is a definable graph. Thus g is a surjectivehomomorphism and by Lemma 2.3(4) χ ( G ) = χ ( G ). Note that if G contains allfinite subgraphs of Sh n ( ω ) then by Proposition 3.2 so does G (maybe for a different n ). Consequently, we may assume that the graph G = ( V, E ) is ∅ -definable in M r .Let D = Dom(Φ) and let h f i (¯ v i ) : i < ω i be an enumeration of all ∅ -definablefunctions to M . Let U = { F i, | ¯ v i | ( b , . . . , b | ¯ v i |− ) : b , . . . , b | ¯ v i |− ∈ Im(Φ) , i < ω } ⊆ M µ,κ ( A ) . Define a surjective map Ψ : U → M by mapping F i, | ¯ x i | ( b , . . . , b | ¯ v i |− ) to f i (Φ − ( b ) , . . . , Φ − ( b | ¯ v i |− )) . Note that Φ is injective so this is well defined.Let Ψ = (Ψ ) r : ( U ) r → M r , V = { a ∈ ( U ) r : Ψ ( a ) ∈ V } and Ψ = Ψ ↾ V : V → V . Let E = Ψ − ( E ), hence G = ( V , E ) is a graph and note that χ ( G ) = χ ( G )by Lemma 2.3(4). Let ν = (reg( κ ) + µ ) < reg( κ ) .Let X = { x i : i < reg( κ ) } be a set of variables as in Remark 5.3. Let t be the set of pairs ( t, ¯ x ), where ¯ x is a sequence of variables from X of length < reg( κ ) and t is a term in L µ,κ with variables contained in ¯ x . Let t be the subsetof t consisting of pairs of the form ( F i, | ¯ v i | ( t , . . . , t | ¯ v i |− ) , ¯ x ), where i < ω , and( t , ¯ x ) , . . . , ( t | ¯ v i |− , ¯ x ) ∈ t . Let t = { (( s , ¯ x ) , . . . , ( s r − , ¯ x r − )) ∈ ( t ) r : ¯ x = . . . =¯ x r − } . We may enumerate t = { ¯ s i (¯ x i ) : i < ν } , where for ease of notation we write(¯ s, ¯ x ) as ¯ s (¯ x ).Since V is covered by the union of { ¯ s i (¯ a ) : ¯ a ∈ A | ¯ x i | } i<ν , V = S i<ν V i , where V i = { ¯ s i (¯ a ) : ¯ a ∈ A | ¯ x i | } ∩ V .By Lemma 2.3(1), assumption (1) and since κ is regular, there exists some i < ν with χ ( G i ) ≥ κ , where G i = ( V i , E ↾ V i × V i ).Set ¯ s = ¯ s i and ¯ x = ¯ x i . Assume, for simplicity, that¯ s (¯ x ) = ( F ,k ( t , (¯ x ) , . . . , t ,k − (¯ x )) , . . . , F r − ,k r − ( t r − , (¯ x ) , . . . , t r − ,k r − (¯ x ))) . Claim. ¯ s defines a surjective function A | ¯ x | → V i .Proof. Since G i is non-empty, there exists ¯ a ∈ A | ¯ x | such that ¯ s (¯ a ) ∈ V . Let ¯ b ∈ A | ¯ x | . By Definition 5.4(1), F i,k i ( t i, (¯ b ) , . . . , t i,k i − (¯ b )) ∈ U for all i < r . Note that | ¯ x | < | A | by assumption (3) and so there exists a permutation π of A mapping ¯ a to ¯ b , and let b π be induced automorphism of M µ,κ . Thus b π (¯ s (¯ a )) = ¯ s (¯ b ). Since V is ∅ -definable and Ψ (¯ s (¯ a )) ∈ V , Definition 5.4(2) gives that tp M (Ψ (¯ s (¯ b )) =tp M (Ψ (¯ s (¯ a )) and hence Ψ (¯ s (¯ b ) ∈ V as well. Consequently, ¯ s defines a function.Surjectivity is straightforward. (cid:3) (claim)Let R = ¯ s − ( E ↾ V i × V i ) be the edge relation ¯ s induces on A | ¯ x | .By assumptions (2 , Let ¯ a, ¯ b, ¯ c, ¯ d ∈ A | ¯ x | satisfying ¯ a R ¯ b and f ¯ a, ¯ b = f ¯ c, ¯ d . The latter condition impliesthat the coordinate-wise map sending ¯ a ¯ b to ¯ c ¯ d is well defined and injective. Since | ¯ x | < | A | , we may find a permutation π of A which maps ¯ a ¯ b to ¯ c ¯ d . This permutationlifts to an automorphism b π of the free algebra, with b π (¯ s (¯ a )) = ¯ s (¯ c ) and b π (¯ s (¯ b )) =¯ s ( ¯ d ).Thus b π ( t i,j (¯ a )) = t i,j (¯ c ) and b π ( t i,j (¯ b )) = t i,j ( ¯ d ), for i < r and j < k i . ByDefinition 5.4(2),tp M ((Φ − ( t i,j (¯ a ))) i We say that a theory T is stationary if all types (over any set)are stationary. Remark . Rothmaler studies stationarity of modules in [Rot83], e.g. he gives acomplete description of stationary abelian groups in [Rot83, Theorem 4(ii)]. Fact 5.9. [Rot83, Lemma 2, Theorem 1] Let T be a stable theory. The followingare equivalent:(1) T is stationary;(2) for any A , every formula which is almost over A is over A ;(3) all -types over (over any set) are stationary. Proposition 5.10. Let T be a complete stationary stable theory in a language L .For every sublanguage L ⊆ L there is some L ⊆ L ′ ⊆ L with | L ′ | = |L | + ℵ such that T ↾ L ′ is stationary.Proof. Recall that a formula ϕ ( x, d ) is almost over A if there exists an equivalencerelation with finitely many classes E ( x, x ′ ) over A such that ∀ x ∀ x ′ ( E ( x, x ′ ) → ( ϕ ( x, d ) ↔ ϕ ( x ′ , d ))). Equivalently, for every equivalence relation with finitelymany classes E ( x, x ′ ) over A every class of E is definable over A . Claim. For every ψ ( x, x ′ , z ) and n < ω there are finitely many formulas θ i ( x, z ) ( i < k ) such that ( † ) for any z -tuple c such that ψ ( x, x ′ , c ) defines an equivalence relation with ≤ n classes, and for any x ′ -tuple d there is some i < k such that ψ ( x, d, c ) is equivalent to θ i ( x, c ) . NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 19 Proof. Note that ( † ) is a first order sentence.Suppose not and fix ψ ( x, x ′ , z ) and n < ω . This means that for every finitecollection of formulas θ i ( x, z ) ( i < k ) there are some c and d witnessing the failureof ( † ). Let Γ( x ′ , z ) be { ψ ( − , − , z ) defines an equivalence relation with ≤ n classes }∪{∃ x ¬ ( ψ ( x, x ′ , z ) ↔ θ ( x, z )) : θ ( x, z ) any formula } . By assumption, Γ is consistent. Let ( d, c ) | = Γ( x ′ , z ). Then ψ ( x, d, c ) is almost over c but not over c , contradiction. (cid:3) (claim)Now, let L ⊆ L be a sublanguage. We construct an increasing sequence oflanguages L m as follows. The language L is given. Assume we have constructed L m . For any ψ ( x, x ′ , z ) in the language L m and n < ω let { θ ψ,n,i ( x, z ) } i Lemma 5.11. If T is a complete stationary stable theory then dcl( A ) = acl( A ) forany set A .Remark . If T is a stable theory, then, since for any A every type over acl eq ( A )is stationary, if T is eliminates imaginaries and has no algebraicity (i.e. acl( A ) =dcl( A ) for any A ) then T is stationary. However, as the theory of the infinite setshows, the other direction is not true (it does not eliminate imaginaries).We will need the following lemma, which is a consequence of [She78, LemmaIII.3.10], but for the convenience of the reader we give a direct proof. Recall thedefinition of κ ( T ) from [She78, Definition III.3.1]. For any infinite indiscerniblesequence I and a set A , let lim( I/A ) be the limit type of I in A (it is denoted byAv( I, A ) in [She78]), i.e.lim( I/A ) = { ϕ ( x, c ) : c ∈ A, ϕ ( a, c ) holds for cofinitely many a ∈ I } . It is a consistent complete type over A by stability. Lemma 5.13. Let T be a stationary stable theory, M a model and λ > κ ( T ) acardinal. If for every non-algebraic type q ∈ S ( C ) with | C | < κ ( T ) and C ⊆ M there is a C -independent set of realizations of q in M of cardinality λ , then M is λ -saturated.Proof. Let p ∈ S ( A ) be a complete type with | A | < λ . If p is algebraic then it isrealized, so we may assume that p is non-algebraic. Let C ⊆ A with | C | < κ ( T ) besuch that p does not fork over C . By assumption, we may find a C -independent set I of realizations of p | C in M (so indiscernible over C by stationarity). By [She78,Lemma III.1.10(2)], lim( I/A ) = p . By [She78, Corollary III.3.5(1)], there is I ⊆ I with I \ I indiscernible over A and | I | ≤ κ ( T )+ | A | < λ . In particular, | I \ I | ≥ ℵ and thus for every c ∈ I \ I , p = tp( c/A ). (cid:3) We fix the following notation for the rest of the section. Let T be completestable theory, and let U be a monster model. Let κ = κ r ( T ), i.e. κ = κ ( T ) + if κ ( T ) is singular or κ ( T ) if not (for the sake of the following, one can also take κ = | T | + ) and let µ = µ <κ be a cardinal, with µ > κ , such that T is µ -stable, e.g.if µ ≥ | T | (see [She78, Lemma III.3.6]), and thus there exists a saturated modelof cardinality µ [She78, Theorem III.3.12]. Fix some partition µ = ∪· i<κ U i to setseach of cardinality µ . From now on we also assume that T is stationary. Definition 5.14. Let I be any set. We define OB ( I ) to be the collection of triples a := (cid:0) i a , { U a j } j
Let I be a set and OB ( I ) as above.(1) We say that a ∈ OB ( I ) is homogeneous if for any permutation π of I , theset of pairs π [ a ] := { ( b a α,η , b a α, b π ( η ) ) : ( α, η ) ∈ W a
Let I be any set and a ∈ OB ( I ) with i a = κ . For every C ⊆ B a <κ with | C | < κ ( T ) ≤ κ there exist some j < κ and η ∈ I j satisfying C ⊆ { b α,ν ∈ B a Since κ is regular, there exist ˜ j < κ with C ⊆ B a < ˜ j . Let J = S b α,ν ∈ C Range( ν ).Let j = max { ˜ j, | J |} < κ and η ∈ I j with Range( η ) = J . (cid:3) Proposition 5.17. Let I be any set with | I | ≥ µ . If a ∈ OB ( I ) is full and i a = κ then M := U ↾ dcl( B a <κ ) is a saturated elementary substructure of U of cardinality | I | <κ .Proof. To show that it is an elementary substructure we use Tarski-Vaught. Let ϕ ( x, b ) be a consistent formula with b ∈ dcl( B a <κ ). There is no harm in assumingthat b ∈ B a <κ . If ϕ ( x, b ) is algebraic then by Lemma 5.11, any realization is alreadyin dcl( B b <κ ). Otherwise, let p be any non-algebraic complete type over b containing ϕ ( x, b ). Let j < κ and η ∈ I j be given by Lemma 5.16 for C = { b } . By stationarity,there is a unique non forking extension p of p to B a By assumption of fullness, for every ν ∈ ∆ there is some α ν ∈ U a lg ( ν ) , suchthat b α ν ,ν | = q | B a Let I be a set. For a , b ∈ OB ( I ) we say that a ≤ b if(1) i a ≤ i b ;(2) for j < i a we have U a j = U b j and b a α,η = b b α,η , for ( α, η ) ∈ W a j . Proposition 5.19. Let I be a set with | I | ≥ κ . Then there exists a full homogeneous a ∈ OB ( I ) with i a = κ .Proof. We choose full homogeneous a j ∈ OB ( I ) by induction on j ≤ κ such that a j ∈ OB ( I ) with i a j = j and such that k ≤ k < j implies a k ≤ a k .For j = 0 choose a = (0 , ∅ , ∅ ) and note that the conditions hold trivially.Let j ≤ κ be a limit ordinal, set i a j = j , U a j k := U a k +1 k for k < j and B a j := S k 2. By the inductionhypothesis b α ,η . . . b α k − ,η k − B b Theorem 5.20. Let T be a complete stationary stable theory. Let κ = κ r ( T ) and M | = T a saturated model of cardinality ≥ µ = µ <κ such that µ ≥ | T | (so T is µ -stable) and κ < µ . Let I be any set such that | I | <κ = | M | .Then there exists a skeletal homogeneous representation of M in M µ,κ ( I ) . Infact, the representation will be in M µ,κ, ( I ) .Proof. Let I be any set such that | I | <κ = | M | ≥ µ and let a ∈ OB ( I ) be a fullhomogeneous object with i a = κ as supplied by Proposition 5.19. By Proposition5.17, dcl( B a <κ ) | = T is saturated of cardinality | I | <κ = | M | . In particular M isisomorphic to dcl( B a <κ ). Without loss of generality we assume M = dcl( B a <κ ). Wedefine a function Φ : B a <κ → M µ,κ, ( I )by Φ( b α,η ) = F α,j ( η ) for the unique j < κ such that η ∈ I j and α ∈ U a j .The map Φ is injective. If F α,j ( η ) = F β,k ( ν ) then, since it is a free algebra, α = β, j = k and η = ν so b α,η = b β,ν .The map Φ is a homogeneous representation. Indeed, to show condition (1), let t (¯ x ) be any term with | ¯ x | = β < κ and let ¯ a ∈ I β with t (¯ a ) ∈ Im(Φ). Thus thereexist j < κ and α ∈ U a j such that t (¯ a ) = F α,j ( η ¯ a ) for some η ¯ a ∈ I j . Thus for any NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 23 ¯ b ∈ I β there is some η ¯ b ∈ I j with t (¯ b ) = F α,j ( η ¯ b ). In particular, t (¯ b ) ∈ Im(Φ), asneeded.For condition (2), let b α ,η . . . b α k ,η k ∈ ( B a <κ ) k and let π be a permutationof I . Since a is homogeneous tp( b α ,η . . . b α k ,η k ) = tp( b α , b π ( η ) . . . b α k , b π ( η k ) ), asneeded. (cid:3) Corollary 5.21. Let G = ( V, E ) be a graph that is interpretable (possibly withparamters) in a stationary stable structure. If χ ( G ) > i ( ℵ ) then there exists an n ∈ N such that G contains all finite subgraphs of Sh n ( ω ) .Proof. Assume G is interpretable in a stationary stable structure N over somefinite set of a parameters A ⊆ N and let T = T h ( N ). Since adding constants to thelanguage preserves stationarity, we may assume that G is interpretable in N over ∅ . Since the interpretation only uses a finite fragment of the language, by applyingProposition 5.10 we may assume that | T | = ℵ .Let µ = 2 ℵ and κ = κ ( T ). Note that κ ( T ) ≤ ℵ ([She78, Corollary III.3.3])which implies κ ( T ) = κ r ( T ) and µ <κ = µ . Let I be any set satisfying | I | ≥ max { µ, | N |} (which implies | I | <κ ≥ max { µ, | N |} ) and let M | = T be a saturatedelementary extension of N of cardinality | I | <κ (exists by [She78, Lemma III.3.6and Theorem III.3.12]).By Theorem 5.20 and Proposition 5.6, there exists n ∈ N such that ( G ( M ) , E ( M )),the realizations in M of the interpretation of G , contains all finite subgraphs ofSh n ( ω ) and since N ≺ M the result follows. (cid:3) Quantitative Bounds The following section is joint work with Elad Levi.The aim of this section is to prove that if G = ( V, E ) is an ω -stable graph withuncountable chromatic number and the U-rank of G , U( G ) , is at most 2 then itcontains all finite subgraphs of Sh n ( ω ) for some n ≤ 2. For the definition of U-ranksee [TZ12, Definition 8.6.1]. Throughout, we will use Lascar’s equality when theU-rank is finite, see [TZ12, Exercise 8.6.5].For certain parts of the argument we will need the following assumption. Assumption ♦ . G is a saturated ω -stable structure, with home sort V , that elim-inates imaginaries in a countable language with acl( ∅ ) = dcl( ∅ ) . Let E ⊆ V be an ∅ -type-definable set. Let p ∈ S ( ∅ ) be a non-algebraic and of finite U -rank such that G p = ( p ( G ) , E ↾ p ( G )) is a graph with χ ( G p ) ≥ ℵ . Note that assumption ♦ implies that every type over ∅ is stationary.Assume ♦ and let E alg = { ( a, b ) ∈ E : a ∈ acl( b ) ∧ b ∈ acl( a ) } be the set ofinteralgebraic pairs belonging to E . Note that if U( a ) = U( b ) then a ∈ acl( b ) if andonly if b ∈ acl( a ). Indeed, by Lascar’s equalityU( a/b ) + U( b ) = U( ab ) = U( a ) + U( b/a )and for any type q , U( q ) = 0 if and only if it is algebraic, see [TZ12, Exercise 8.6.1].Let E nalg = E \ E alg , it is definable by a countable type. Lemma 6.1. Assume ♦ .(1) χ ( p ( G ) , E nalg ) ≥ ℵ ;(2) If there exist a, b ∈ G p with a E b and a | ⌣ b then any Morley sequencebased on p forms an infinite complete graph;(3) If U( p ) = 1 then there exists a Morley sequence based on p which forms aninfinite complete graph Proof. (1) By interalgebraicity, every connected component of ( p ( G ) , E alg ) is count-able and consequently χ ( p ( G ) , E alg ) ≤ ℵ . By Lemma 2.3(2), χ ( p ( G ) , E nalg ) ≥ ℵ .(2) Assume there exist a, b ∈ G p with a E b and a | ⌣ b . Since every type over ∅ is stationary it follows that every Morley sequence based on p forms an infinitecomplete graph.(3) Assume U( p ) = 1. Since χ ( p ( G ) , E nalg ) ≥ ℵ , there must exist some a, b ∈ p ( G ) with a E nalg b . If a ⌣ b then U( a/b ) < U( a ) = 1, which implies that a ∈ acl( b )(so b ∈ acl( a ))), contradiction. Thus a | ⌣ b and we may use (1). (cid:3) Definition 6.2. We say that a stationary type tp( a/A ) is pseudo-one-based ifCb( a/A ) ⊆ acl eq ( a ). Remark . Compare with the last paragraph of page 105 in [Pil96].We give some examples of pseudo-one-based types. Lemma 6.4. (1) In a one-based theory every stationary type (over any base)is pseudo-one-based.(2) Let M be a stable structure. If U( a ) = U( b ) = 1 and a ⌣ b then tp( a/ acl eq ( b )) is pseudo-one-based.(3) Let M be a stable structure and a, b ∈ M non interalgebraic, with U( a ) =U( b ) = 2 . Let X and Y be infinite mutually indiscernible sets with a ∈ X and b ∈ Y . If a ⌣ b then tp( a/ acl eq ( b )) is pseudo-one-based.Proof. (1) A stable theory is one-based if for all a, B , Cb( a/ acl eq ( B )) ⊆ acl eq ( a ).Note that since tp( a/A ) is stationary, Cb( a/ acl eq ( A )) = Cb( a/A ). The resultfollows.(2) Since a ⌣ b , by U-rank considerations as before, a and b are interalgebraic.So Cb( a/ acl eq ( b )) ⊆ acl eq ( b ) ⊆ acl eq ( a ).(3) We note that for any b = b ′ ∈ Y , a | ⌣ b ′ b . Indeed, otherwise U( a/bb ′ ) < U( a/b ′ ) < U( a ) and since U( a ) = 2, a ∈ acl( bb ′ ), contradicting the mutual indis-cernibility of X, Y . Similarly, a | ⌣ b b ′ . Consequently, setting e := Cb( a/ acl eq ( bb ′ )), e ⊆ acl eq ( b ) ∩ acl eq ( b ′ ) and a | ⌣ e bb ′ . Similarly we get that b | ⌣ a b ′ and by theproperties of forking e | ⌣ a e and hence e ∈ acl eq ( a ). Finally, since a | ⌣ e b (and e ∈ acl eq ( b )), Cb( a/ acl eq ( b )) ⊆ acl eq ( e ) ⊆ acl eq ( a ), as needed. (cid:3) Abundance of pseudo-one-based types will be a key tool in our proofs. Theabove shows that this can be achieved in one-based theories and U-rank 1 types.For U-rank 2 we observe the following: Lemma 6.5. Assume ♦ and that U( p ) = 2 . Then either we can embed an infinitecomplete graph into G p or there exists a type-definable symmetric irreflexive relation E ⊆ E nalg such that ( † ) for every ( a, b ) ∈ E , tp( a/ acl( b )) is pseudo-one-based. Moreover, if F ⊆ E is a symmetric irreflexive type-definable relation with χ ( G p , F ) ≥ ℵ then F ∩ E = ∅ .Proof. Assume that we cannot embed an infinite complete graph into G p , in par-ticular by Lemma 6.1, for every a, b | = p with a E b , a ⌣ b .Let E be the set of pairs ( a, b ) ∈ E ↾ p ( G ) such that there exists a completebipartite subgraph K X,Y of G p such that X and Y are infinite mutually indiscerniblesets with a ∈ X and b ∈ Y . Easily, E is type-definable by a countable type. Since,by [EH66, Corollary 5.6], G p contains K n,n (the complete bipartite graph on n vertices) for every n < ω , E non-empty. Clearly, E ⊆ E nalg . By Lemma 6.4(3),for every ( a, b ) ∈ E , tp( a/ acl( b )) is pseudo-one-based.For the moreover part, if χ ( G p , F ) ≥ ℵ then by by [EH66, Corollary 5.6] we mayembed K n,n into it for any n < ω . Thus by saturation necessarily F ∩ E = ∅ . (cid:3) NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 25 The following is the key proposition of the proof and where pseudo-one-basedtypes show their usefulness. Proposition 6.6. Assume ♦ and that E ⊆ E nalg is a type-definable symmetricirreflexive relation satisfying ( † ) from Lemma 6.5.(1) For any ( a, b ) ∈ E there is a finite tuple e such that a | ⌣ e b and tp( a/e ) isstationary.(2) Let Ψ be the collection of all pairs of formulas ( ϕ ( u, x ) , ψ ( u, x )) satisfying(a) ϕ ( u, a ) and ψ ( u, a ) are algebraic formulas each isolating a completetype over some (any) a | = p ;(b) there exist a, b | = p and e such that ( a, b ) ∈ E , a | ⌣ e b , ϕ ( e, a ) , ψ ( e, b ) and tp( a/e ) is stationary.For any ( ϕ, ψ ) ∈ Ψ let E ϕ,ψ = { ( a, b ) ∈ E nalg : a, b ∈ p ( G ) , ∃ e ( ϕ ( e, a ) ∧ ψ ( e, b )) } . Then either ( ∗ ) we can embed an infinite complete graph into ( p ( G ) , E nalg ) or there exists ( ϕ ( u, x ) , ψ ( u, x )) ∈ Ψ such that ( ∗∗ ) ϕ,ψ : p ⊢ ∀ u ( ϕ ( u, x ) →¬ ψ ( u, x )) and χ ( G p,ϕ,ψ ) ≥ ℵ , where G p,ϕ,ψ := ( p ( G ) , E { ϕ,ψ } ) and E { ϕ,ψ } = E ϕ,ψ ∨ E ψ,ϕ .(3) Assume ( ∗∗ ) ϕ,ψ . There exists q ϕ ∈ S ( ∅ ) such that for any a | = p and e | = ϕ ( u, a ) , e | = q ϕ . Similarly, there exists q ψ ∈ S ( ∅ ) such that for any a | = p and e | = ψ ( u, a ) , e | = q ψ . Furthermore, q := q ϕ = q ψ and < U( q ) < U( p ) .(4) Assume ( ∗∗ ) ϕ,ψ and let q = q ϕ = q ψ . The type-definable relation e R e given by ( ∃ a | = p ) (( ϕ ( e , a ) ∧ ψ ( e , a ) ∨ ( ϕ ( e , a ) ∧ ψ ( e , a ))) , defines a graph H q on realizations of q with χ ( H q ) ≥ ℵ .Proof. We start by showing that for ( a, b ) ∈ E , E ⊆ [ ( ϕ,ψ ) ∈ Ψ E { ϕ,ψ } . Let ( a, b ) ∈ E . Since tp( a/ acl( b )) is pseudo-one-based, Cb( a/ acl( b )) ⊆ acl( a ) ∩ acl( b ). By [TZ12, Exercise 8.4.7], there is a finite tuple e such that dcl( e ) =Cb( a/ acl( b )) (this proves (1)). We choose ϕ ( u, a ) to be a formula isolating tp( e/a )and ψ ( u, b ) to be a formula isolating tp( e/b ). Hence ( a, b ) ∈ E { ϕ,ψ } and ( ϕ, ψ ) ∈ Ψ.Since E is type-definable, by saturation there exists a finite subset Ψ ⊆ Ψ suchthat E ⊆ [ ( ϕ,ψ ) ∈ Ψ E { ϕ,ψ } . Assume that we cannot embed an infinite complete graph into ( p ( G ) , E nalg ). Claim. For any ( ϕ, ψ ) ∈ Ψ , p ⊢ ∀ u ( ϕ ( u, x ) → ¬ ψ ( u, x )) .Proof. Choose any ϕ, ψ ∈ Ψ and let ( a, b ) ∈ E and e be such that ϕ ( e, a ), ψ ( e, b ), a | ⌣ e b and tp( a/e ) stationary. Assume, toward a contradiction that there is some e ′ with ϕ ( e ′ , b ) and ψ ( e ′ , b ). Since both ϕ ( u, b ) and ψ ( u, b ) isolate a complete typeover b and are mutually consistent, they must be equivalent (i.e. define the samedefinable set). So ϕ ( e, a ) and ϕ ( e, b ).Let σ be an automorphism satisfying σ ( a ) = b . Applying to the formulas above: ϕ ( e, b ) and ϕ ( σ ( e ) , b ). Since ϕ ( u, b ) isolates a complete type over b there exists anautomorphism τ fixing b and mapping σ ( e ) to e . Combining, τ ◦ σ fixes e and maps a to b , i.e. a ≡ e b . By assumption tp( a/e ) is stationary and b | = tp( a/e ) | ea so we may construct aMorley sequence over e starting with a, b . Since a E b we get an infinite completegraph, contradicting our assumption. (cid:3) (claim)Note that each E { ϕ,ψ } defines a graph relation. Let θ ( x, y ) = _ ( ϕ,ψ ) ∈ Ψ ∃ e ( ϕ ( e, x ) ∧ ψ ( e, y )) ∨ ∃ e ( ϕ ( e, y ) ∧ ψ ( e, x )) . Set E = { ( a, b ) ∈ E nalg : θ ( a, b ) } and E = { ( a, b ) ∈ E nalg : ¬ θ ( a, b ) } . Obviously, E nalg = E ∪ E and both E and E are symmetric. If χ ( p ( G ) , E ) ≥ ℵ then by( † ) there exists ( a, b ) ∈ E ∩ E , contradicting the choice of Ψ . Thus, by Lemma2.3(2), χ ( p ( G ) , E ) ≥ ℵ . Again by Lemma 2.3(2), there is some ( ϕ, ψ ) ∈ Ψ suchthat χ ( G p,ϕ,ψ ) ≥ ℵ .(3) Let q ϕ = tp( e ) for some e | = ϕ ( u, a ) and some a | = p and let q ψ = tp( e )for some e | = ψ ( u, b ) and some b | = p . Since p is a complete type and ϕ and ψ each isolate a complete type it follows that q ϕ and q ψ do not depend on a , b , e or e .As ( ϕ, ψ ) ∈ Ψ, there exist ( a, b ) ∈ E nalg and e such that ϕ ( e, a ) ∧ ψ ( e, b ) and a | ⌣ e b . Consequently, e | = q ϕ and e | = q ψ and hence q ϕ = q ψ . Since e ∈ acl( a ),U( a/e ) + U( e ) = U( a ) . If U( e ) = 0 then a | ⌣ e so by transitivity of forking a | ⌣ b , but then wemay embed an infinite complete graph as in Lemma 6.1, which contradicts (2). IfU( a/e ) = 0 then a ∈ acl( e ) ⊆ acl( b ) so a and b are interalgebraic, contradiction(see above Lemma 6.1).(4) Note that R defines a graph, i.e. it is irreflexive by (2). Let n be | ϕ ( G, a ) | and m be | ψ ( G, a ) | for some (any) a | = p . For any a | = p choose enumerations ϕ ( G, a ) = { e i ( a ) : i < n } and ψ ( G, a ) = { e ′ i ( a ) : i < m } .For i < n and j < m let H i,j = { ( e i ( a ) , e ′ j ( a )) : a | = p } . We define an edge relation on H i,j = { ( e i ( a ) , e ′ j ( a )) : a | = p } (for i < n, j < m )as follows: ( e i ( a ) , e ′ j ( a )) is connected to ( e i ( b ) , e ′ j ( b )) if and only if e i ( b ) = e ′ j ( a ) or e i ( a ) = e ′ j ( b ). Note that e i ( a ) = e ′ j ( a ) for all a | = p and i < n, j < m by (2), hencethis relation is irreflexive. Claim. There exist i < n and j < m such that χ ( H i ,j ) ≥ ℵ .Proof. Assume that for all i < n, j < m , H i,j is countably colorable, say by the col-oring function c i,j : H i,j → ℵ . We claim that this entails that G p,ϕ,ψ is countablycolorable, which would give a contradiction to choice of ( ϕ, ψ ).We define a coloring c : G p,ϕ,ψ → ( ℵ ) n × m by c ( a )( i, j ) = c i,j ( e i ( a ) , e ′ j ( a )).The contradiction will follow if we show that this is a legal coloring. Let ( a, b ) ∈ E ϕ,ψ (( a, b ) ∈ E ψ,ϕ is similar). Thus there exists some e | = ϕ ( u, a ) ∧ ψ ( u, b ).Consequently, e = e i ( a ) = e ′ j ( b ) for some i < n, j < m , so c i,j ( e i ( a ) , e ′ j ( a )) = c i,j ( e i ( b ) , e ′ j ( b )) , and c ( a ) = c ( b ). (cid:3) (claim)Now, assume that χ ( H q ) ≤ ℵ and let c : q ( G ) → ℵ be a coloring. We define f : H i ,j → ℵ × ℵ by f ( e i ( a ) , e ′ j ( a )) = (cid:0) c ( e i ( a )) , c ( e ′ j ( a )) (cid:1) . This gives alegal coloring of H i ,j using countably many colors, and we reach a contradiction:assume with out loss of generality that ( e i ( a ) , e ′ j ( a )) , ( e i ( b ) , e ′ j ( b )) ∈ H i ,j with e ′ j ( a ) = e i ( b ). Since e i ( a ) R e ′ j ( a ), c ( e i ( a )) = c ( e ′ j ( a )) = c ( e i ( b )) and thus f ( e i ( a ) , e ′ j ( a )) = f ( e i ( b ) , e ′ j ( b )), as needed. (cid:3) NFINITE STABLE GRAPHS WITH LARGE CHROMATIC NUMBER 27 Remark . We remark that if tp( a/e ) is stationary and e ′ | = tp( e/a ) then tp( a/e ′ )is also stationary.The procedure outlined in the items of Proposition 6.6 supplies, under someassumptions, a graph, with uncountable chromatic number, concentrated on a typeof lower U-rank than the one we started with. This hints that some inductionprocedure may be possible (at least for one-based theories). We will not pursuethis further now. For now we concentrate on graphs of at most U-rank 2.The following is an easy exercise in stability theory. Lemma 6.8. Let T be a stable theory. Assume that A | ⌣ C B , A ′ | ⌣ C B ′ , B ≡ C B ′ , A ≡ C A ′ and tp( A/C ) stationary. Then AB ≡ C A ′ B ′ . Theorem 6.9. Let G = ( V, E ) be an ω -stable graph with χ ( G ) ≥ ℵ . If U( G ) ≤ then G contains all finite subgraphs of Sh n ( ω ) for some n ≤ .Proof. We may assume that G is saturated (in particular ℵ -saturated) and we mayalso work in G eq . Fix some countable G ≺ G and add constants for it (so everytype over ∅ is stationary). By ω -stability and Lemma 2.3(1) there is some type p ∈ S ( ∅ ) such that χ ( G p ) ≥ ℵ with G p = ( p ( G ) , E ↾ p ( G )). We are now in thesituation of Assumption ♦ .If U( p ) = 0 then p is algebraic (even realized), contradicting χ ( G p ) ≥ ℵ . IfU( p ) = 1 then we may embed an infinite complete graph by Lemma 6.1(3).We may thus assume that U( p ) = 2 and that G does not contain an infinitecomplete graph.Let E ⊆ E be the type-definable set from Lemma 6.5 and ϕ , ψ, G p,ϕ,ψ , q , R and H q be as supplied by Proposition 6.6 with respect to E and p (so necessarilyU( q ) = 1). Noting that Assumption ♦ is true for H q , we may apply Lemma 6.1(3)and thus there exist a Morley sequence h e i : i < ω i such that e i R e j for all i = j .Since e R e then there is some a , | = p such that, without loss of generality, ϕ ( e , a , ) ∧ ψ ( e , a , ). For any i < j < ω let a i,j be such that a i,j e i e j ≡ a , e e .Note that for every i < j < ω , a i,j ∈ acl( e i , e j ). Indeed, by Lascar’s equalityU( a i,j /e i e j ) + U( e i /e j ) = U( a i,j e i /e j ) and since e i ∈ acl( a i,j ) the right hand sideis also equal to U( a i,j /e j ). Now we note that U( a i,j /e j ) + U( e j ) = U( a i,j e j ), butas before e j ∈ acl( a i,j ) so the right hand side is equal to 2 and since U( e j ) = 1 weconclude that U( a i,j /e j ) = 1. As e i | ⌣ e j we have that U( e i /e j ) = 1 as well so wecombine everything and get that U( a i,j /e i e j ) = 0.Define a map f : Sh ( ω ) → G p,ϕ,ψ by ( i, j ) a i,j . We claim that this is aninjective graph homomorphism.For ( i, j ) = ( i ′ , j ′ ) ∈ Sh ( ω ), a i,j and a i ′ ,j ′ are not interalgebraic and in par-ticular f is injective. Indeed, assume acl( a i,j ) = acl( a i ′ ,j ′ ). Since ( i, j ) = ( i ′ , j ′ ), |{ i, j, i ′ , j ′ }| ≥ 3, and we assume that i = j, i ′ , j ′ (the other cases are similar). Onthe other hand, e i ∈ acl( a i,j ) = acl( a i ′ ,j ′ ) ⊆ acl( e i ′ , e j ′ ) , contradicting indiscernibility. f is a graph homomorphism. Let ( i, j ) , ( j, k ) ∈ Sh ( ω ), so i < j < k < ω . As a i,j and a j,k are not interalgebraic, necessarily a i,j | ⌣ e j a j,k for otherwiseU( a i,j /e j a j,k ) < U( a i,j /e j ) = 1and then a i,j ∈ acl( e j a j,k ) ⊆ acl( a j,k ).By the choice of ( ϕ, ψ ) in Proposition 6.6, we may find a, b | = p and e | = q with a E b , a | ⌣ e b , tp( a/e ) stationary, ϕ ( e, a ) and ψ ( e, b ). By applying an au-tomorphsim mapping e j to e we may assume e j = e . Let σ be an automorphismmapping a to a j,k , thus ae ≡ a j,k σ ( e ) and σ ( e ) | = ϕ ( u, a j,k ). Applying now anautomorphism mapping σ ( e ) to e but fixing a j,k we conclude that a ≡ e a j,k and similarly b ≡ e a i,j . Since tp( a/e ) is stationary, by Lemma 6.8, ab ≡ a j,k a i,j so a i,j E a j,k as well. (cid:3) References [CPT20] G. CONANT, A. PILLAY, and C. TERRY. A group version of stable regularity. Math.Proc. Cambridge Philos. Soc. , 168(2):405–413, 2020.[CS16] Moran Cohen and Saharon Shelah. Stable theories and representation over sets. MLQMath. Log. Q. , 62(3):140–154, 2016.[EH66] P. Erd˝os and A. Hajnal. On chromatic number of graphs and set-systems. Acta Math.Acad. Sci. Hungar. , 17:61–99, 1966.[EH68] P. Erd˝os and A. Hajnal. On chromatic number of infinite graphs. In Theory of Graphs(Proc. Colloq., Tihany, 1966) , pages 83–98. Academic Press, New York, 1968.[EHS74] P. Erd˝os, A. Hajnal, and S. Shelah. On some general properties of chromatic numbers.In Topics in topology (Proc. Colloq., Keszthely, 1972) , pages 243–255. Colloq. Math.Soc. J´anos Bolyai, Vol. 8, 1974.[ER50] P. Erd¨os and R. Rado. A combinatorial theorem. J. London Math. Soc. , 25:249–255,1950.[HK84] Andr´as Hajnal and P´eter Komj´ath. What must and what need not be contained in agraph of uncountable chromatic number? Combinatorica , 4(1):47–52, 1984.[Hod93] Wilfrid Hodges. Model theory , volume 42 of Encyclopedia of Mathematics and its Ap-plications . Cambridge University Press, Cambridge, 1993.[Kom11] P´eter Komj´ath. The chromatic number of infinite graphs—a survey. Discrete Math. ,311(15):1448–1450, 2011.[KS05] P´eter Komj´ath and Saharon Shelah. Finite subgraphs of uncountably chromatic graphs. J. Graph Theory , 49(1):28–38, 2005.[Mar99] Benoit Mariou. Modeles engendres par des indiscernables et modeles satures . PhD thesis,1999. Th`ese de doctorat dirig´ee par Bouscaren, ´Elisabeth Logique et fondements del’informatique Paris 7 1999.[Mar01] Benoˆıt Mariou. Mod`eles satur´es et mod`eles engendr´es par des indiscernables. J. SymbolicLogic , 66(1):325–348, 2001.[MS14] M. Malliaris and S. Shelah. Regularity lemmas for stable graphs. Trans. Amer. Math.Soc. , 366(3):1551–1585, 2014.[Pil96] Anand Pillay. Geometric stability theory , volume 32 of Oxford Logic Guides . The Claren-don Press, Oxford University Press, New York, 1996. Oxford Science Publications.[PZ78] Klaus-Peter Podewski and Martin Ziegler. Stable graphs. Fund. Math. , 100(2):101–107,1978.[Rot83] Philipp Rothmaler. Stationary types in modules. Z. Math. Logik Grundlag. Math. ,29(5):445–464, 1983.[She] Saharon Shelah. Divide and Conquer: Dividing lines on universality.[She78] Saharon Shelah. Classification theory and the number of nonisomorphic models , vol-ume 92 of Studies in Logic and the Foundations of Mathematics . North-Holland Pub-lishing Co., Amsterdam-New York, 1978.[She19] Saharon Shelah. Superstable theories and representation. preprint, https://arxiv.org/abs/1412.0421 , 2019.[Tay70] Combinatorial structures and their applications , volume 1969 of Proceedings of the Cal-gary International Conference on Combinatorial Structures and their Applications heldat the University of Calgary, Calgary, Alberta, Canada, June . Gordon and Breach,Science Publishers, New York-London-Paris, 1970.[Tay71] Walter Taylor. Atomic compactness and elementary equivalence. Fund. Math. ,71(2):103–112, 1971.[Tho83] Carsten Thomassen. Cycles in graphs of uncountable chromatic number. Combinatorica ,3(1):133–134, 1983.[TZ12] Katrin Tent and Martin Ziegler. A course in model theory , volume 40 of Lecture Notesin Logic . Association for Symbolic Logic, La Jolla, CA; Cambridge University Press,Cambridge, 2012. E-mail address : [email protected] E-mail address : [email protected] E-mail address ::