Chromatic number of infinite graphs

Transcription

1 Chromatic number of infinite graphs Jerusalem, October 2015

2 Introduction [S] κ = {x S : x = κ} [S] <κ = {x S : x < κ}. A graph is (V,X) or simply X where V is the set of vertices, X [V] 2 is the set of edges. N(x) = {y : {x, y} X }, N (x) = {y < x : {x, y} X }, d (x) = N (x)

3 Introduction A good coloring is a function f : V µ such that if {x, y} X then f (x) f (y). The chromatic number of X, Chr(X) is the minimal µ such that a good coloring f : V µ exists.

4 Introduction Theorem. (Galvin K) The statement that each graph has a chromatic number is equivalent to the Axiom of Choice.

5 de Bruijn Erdős theorem Theorem. (de Bruijn Erdős) If n < ω, X is a graph, for every finite W V, Chr(X W) n, then Chr(X) n.

6 de Bruijn Erdős theorem Proof. (1) Ultrafilter. Let U be an ultrafilter on [V] <ω such that {s [V] <ω : x s} U (x V). For each s [V] <ω let f s : s n be a good coloring of X s. For x V let g(x) < n be the unique color such that {s : f s (x) = g(x)} U. g : V n is a good coloring: if {x, y} X, then {s : x, y s}, {s : f s (x) = g(x)}, {s : f s (y) = g(y)} are all in U and for any s like that, f s (x) f s (y).

7 de Bruijn Erdős theorem In other words: X can be embedded into the ultraproduct of its finite subgraphs X s (X s)/u. The good colorings of the factors give a good coloring of the ultraproduct.

8 de Bruijn Erdős theorem (2) Gödel s compactness theorem.

9 de Bruijn Erdős theorem Theorem. (Rado s selection principle) Assume that A v is finite for v V, and for every = s [V] <ω there is a family = F s {A v : v s} of functions such that if s t [V] <ω, f F t, then f s F s. Then there is g {A v : v V } such that g s F s for = s [V] <ω.

10 Coloring number The coloring number of a graph (V,X), Col(V,X), is the least cardinal µ such that there is a well ordering < of V such that each v V is joined into less than µ smaller vertices. (A µ-good well ordering.) We have Chr(X) Col(X).

11 Coloring number Examples. If κ, λ are infinite cardinals, (a) Col(K κ ) = κ, (b) Col(K κ,κ ) = κ, (c) if κ < λ, then Col(K κ,λ ) = κ +.

12 Coloring number Theorem. Let µ ω be a cardinal. If X is a graph on the vertex set V, then the following are equivalent. (a) Col(X) µ, (b) there is an ordering < of V such that d (x) < µ (x V), (c) there is a set mapping f : V [V] <µ such that if {x, y} X, then either y f (x) or x f (y),

13 Coloring number Proof. (a) (b) Clear from definition. (b) (c) Direct all edges going down, i.e., f (x) = {y < x : {y, x} X }.

14 Coloring number (c) (a) V = {v α : α < λ}. V α = {v β : β < α} {f (v β ) : β < α} {f 2 (v β ) : β < α} Clearly, {V α : α < λ} is increasing, continuous. Set W α = V α+1 V α. Then W α {v α } f (v α ) f 2 (v α ) so W α µ. Also, if x W α, y V α, {x, y} X, then y f (x). Well order W α into µ and make W β < W α (β < α).

15 Coloring number Theorem. If X is a graph on V with V = λ and Col(X) µ, then there is a well ordering of V witnessing Col(X) µ into ordinal λ. Proof. If λ = µ, any well ordering into ordinal will do. If λ > µ, the statement is given by the proof of (c) (a) of the previous Theorem.

16 Coloring number Finite coloring number When the coloring number is finite. De Bruijn Erdős is not true!

17 Coloring number Finite coloring number There is a countable graph X with Col(X) = 4, Col(Y ) 3 for each finite Y X. T n+1 < T n < < T 3 < T 2 < T 1 < T 0 each T n is a triangle. Add 3 edges between T n and T n+1 such that the downdegree of each nodes is 2, each of the 3 edges go into one node of T n+1. Each finite Y has Col(Y ) 3. Assume that X is well ordered. Let max(t n ) be minimal. Then max(t n ) < max(t n+1 ) = x. From x, one edge goes to T n, 2 edges to T n, i.e., d (x) 3.

18 Coloring number Finite coloring number Modifying this, Erdős and Hajnal constructed for 1 n < ω, 2 k < ω a graph X such that X = ℵ n, Col(X) = 2k, and if Y X has Y < ℵ n, then Col(Y ) k + 1. This is sharp by the following Theorem.

19 Coloring number Finite coloring number Theorem. Let X be a graph on some vertex set V, 1 k < ω. Then each of the following statements implies the next. (1) If Y X is finite, then Col(Y ) k + 1. (2) V has a (not necessarily well) ordering in which there at most k edges going down from any vertex. (3) X is the union of k forests. (4) Col(X) 2k.

20 Coloring number Finite coloring number Theorem. If n < ω, X has Col(X) = n + 1, then there is a subgraph Y X, Col(Y ) = n.

21 Coloring number Finite coloring number Theorem. Assume that ω µ < κ = cf(κ), X a graph on κ, all Y X with Y < κ has Col(Y ) µ. Then Col(X) > µ if and only if S(X) = {α < κ : β α, N(β) α µ} is stationary.

22 Coloring number Finite coloring number = : If S(X) is nonstationary, let C be a closed, unbounded set with S(X) C =. We can assume 0 C. C splits κ into complementary intervals [γ, γ ) Each X [γ, γ ) has coloring number µ, re-order it to a well order witnessing it, place them one after the other. This gives a well ordering witnessing Col(X) µ.

23 Coloring number Finite coloring number =: Assume that f : κ [κ] <µ is such that if {α, β} X, then α f (β) or β f (α). Let C κ be a closed, unbounded set, s.t. it is closed under f, i.e., if β < α C, then f (β) α. Pick δ C S(X). Now f (β(δ)) contains N(β(δ)) δ which is of size µ, contradiction!

24 Coloring number Singular card. comp. Theorem. (Singular Cardinal Compactness, Shelah) Assume that λ is singular, µ < λ, (V,X) is a graph with V = λ such that for each A [V] <λ, Col(X A) µ. Then Col(X) µ.

25 Coloring number Singular card. comp. Definition. A V, A < λ is extendable, if N(x) A < µ for every x V A and so every µ-good well ordering of A can be endextended to any B with B < λ.

26 Coloring number Singular card. comp. Lemma. (Erdős-Hajnal) If A V, A κ < λ, then there is an extendable A A, A = κ. Proof. Define the increasing continuous {A α : α < κ + } such that A 0 = A, A α = κ, A α can not be extended to A α+1. If B = {A α : α < κ + }, < is a µ-good well ordering of B into ordinal κ +, then for some α, A α is an initial seqment, contradiction.

27 Coloring number Singular card. comp. Let {λ α : α < cf(λ)} be a continuous, increasing sequence of cardinals, converging to λ, λ 0 > cf(λ), µ. Decompose V as V = {V α : α < cf(λ)}, V α = λ α. The simplest way would be to construct a continuous, increasing sequence {A α : α < cf(λ)} of extandable sets such that A α = λ α, A α V α. But nothing guarantees that the increasing union of extandable sets is extendable, unless cf(λ) cf(µ).

28 Coloring number Singular card. comp. By mathematical induction on n < ω we define for all α simultaneously the extendable A α,n, A α,n = λ α and a µ good well ordering α,n of A α,n such that α,n+1 end-extends α,n. We will have A α = {A α,n : n < ω} with the µ-good well ordering α = { α,n : n < ω}.

29 Coloring number Singular card. comp. (1) V α A α,0 (α < cf(λ)); (2) A α,n+1 A β,n (β α); (3) A α,n is split as {B α,n β : β < α} increasing, continuous, B α,n β = λ β (α is limit); (4) Bα β,n A α,n+1 (β > α); (5) if x, y A α,n, {x, y} X, x α,n y, y A β,n (β < α), then x A β,n+1.

30 Coloring number Singular card. comp. Can be done. (3) immediate, the rest requires that some λ α things be put into A α,n.

31 Coloring number Singular card. comp. (1) V α A α,0 (α < cf(λ)); Guarantees {A α : α < cf(λ)} = V.

32 Coloring number Singular card. comp. (2) A α,n+1 A β,n (β α); Guarantees A α A β (β < α).

33 Coloring number Singular card. comp. (4) B β,n α A α,n+1 (β > α); Guarantees that {A α : α < cf(λ)} is continuous: if α is limit, x A α, x A α,n for some n, then x B α,n β for some β < α, by (4) x A β,n+1 A β.

34 Coloring number Singular card. comp. (5) if x, y A β,n, {x, y} X, x β,n y, y A α,n (α < β), then x A α,n+1. Guarantees that A α is extendable: assume not and y / A α yet it is joined to µ elements of A α. There is some β > α, y A β. As β is a µ-good well order, these µ elements of A α cannot all precede y by β. There are, therefore, n < ω, x A α,n, y A β,n, y β,n x, {x, y} X. But then by (5), y A α,n+1 A α, contradiction.

35 Coloring number Minimal graphs Given µ, we call a graph X of the first kind, and of type (λ, µ), if it is a bipartite graph on the bipartition classes A and B, A = λ, B = λ + for some cardinal λ µ and d(x) = µ for each x B.

36 Coloring number Minimal graphs A graph is a graph of the second kind of type (κ, µ), if it is isomorphic to a (κ, X) where κ is a regular cardinal κ > ω and there is a stationary set S κ such that N (α) is a cofinal subset of α of type µ.

37 Coloring number Minimal graphs Theorem. (a) If X is a graph of first or second kind, of type (λ, µ), then Col(X) > µ. (b) If X is a graph with Col(X) > µ, then X contains a subgraph of the first or second kind of type (λ, µ) for some λ.

38 Coloring number Minimal graphs Proof. (a) Assume that X is a graph of first kind on A B, A = λ, B = λ +, each N(x) = µ (x B). Assume f : A B [A B] <µ is such that if {x, y} X, then either x f (y) or y f (x). If y B {f (x) : x A}, then f (y) N(y) = µ, contradiction. If X is of the second kind, use Fodor.

39 Coloring number Minimal graphs (b) If X is a graph with Col(X) > µ, X = κ, minimal cardinality, so S(X) is stationary in κ. For each α S(X), set f (α) = sup of the first mu elements of N (α). If f (α) < α stat often, find a subgraph of first kind. If f (α) = α stat often, find a subgraph of the second kind.

40 Coloring number Minimal graphs Corollary. (a) (Erdős Hajnal) If Col(X) > ω, then X contains C 4, even every complete bipartite graph K n,m (n, m < ω). (b) (Halin) If Col(X) > κ, then X contains a topological K κ. (c) (Thomassen) If Col(X) > κ, then X contains a κ-edge-connected Y such that Col(Y ) > κ.

41 Coloring number Minimal graphs Proof. (a) Let X be a graph of the first kind, on V = A B, A = λ, B = λ +, N(x) = ω (x B). For {a, b} [A] 2 set N({a, b}) = N(a) N(b). Clearly, {N(s) : s [A] 2, N(s) 1} has size λ. There are, therefore, a, b A, with c, d N({a, b}), but {a, c, b, d} is a C 4. If X is a graph of the second kind, use Fodor s lemma.

42 Coloring number Minimal graphs Incompactness Theorem. If κ > µ ω are regular and there is a nonreflecting stationary set S Sµ κ, then there is a graph X on κ with Col(X) = µ + such that Col(X α) µ for α < κ. Proof. Join each α S into a µ-sequence converging to it.

43 Coloring number Compactness Compactness Lemma. If X is a graph with Col(X) > ω, then Col(X) > ω will stay after forcing with an ω 1 -closed (or just proper) forcing. Proof. Let (P, ) be the forcing. (a) Let X be a graph of the first kind on A B, A = λ, B = λ +, N(x) = ω (x B).

44 Coloring number Compactness Assume that p P forces that f : A B [A B] <ω is such that if {x, y} X, then either x f (y) or y f (x). Let M be an elementary submodel (of some H(θ)) such that p, P, X, f M, M = λ. Pick y B M, N(y) = {x 0, x 1,... }, then select p p 0 p 1 such that p i forces f (x i ) = s i for some finite s i B. If p p i (i < ω) then p forces that {x 0, x 1,... } f (y). (b) X second kind, proof similar, have M such that δ = sup(m λ) S.

45 Coloring number Compactness Theorem. (GCH) If κ is supercompact, then after forcing with Coll(ω 1, < κ) each X with Col(X) > ω contains a Y with Y = Col(Y ) = ω 1.

46 Coloring number Compactness Proof. Set P = Coll(ω 1, < κ). Let G be V P-generic. Let λ = X. Pick an elementary embedding j : V M, crit(j) = κ, j(κ) > λ, [M] λ M. j(p) = P Q where Q = Coll(ω 1, [κ, j(κ)0). Pick a V[G] Q-generic H.

47 Coloring number Compactness As P V κ we can elevate j to j : V[G] M[G, H] by j(τ G ) = j(τ) G,H. By the Lemma, Col(X) > ω holds in V[G, H]. As V[G, H] = [M[G, H]] κ M[G, H], we have M[G, H] = Col(X) > ω.

48 Coloring number Compactness Set Y = j[x] M[G, H] (as j λ M). In M[G, H], there is a subgraph Y of j(x), Y < j(κ), Col(Y ) > ω. Apply j backwards: in V[G], there is Y X, Col(Y ) > ω, Y < κ = ℵ 2.

49 Coloring number Compactness Theorem. (Shelah) If the existence of a proper class of supercompact cardinals is consistent, then it is consistent that if Col(X) > µ, then X contains a graph Y with Y = Col(Y ) = µ +. Proof. Let {κ α : α ORD} be an increasing, continuous equence of cardinals, κ 0 = ω, if κ α is regular, then κ α+1 is supercompact, if κ α is singular, then κ α+1 = κ + α. Iterate such that if κ α is regular then Q α = Coll(κ α, < κ α+1 ), if κ α is singular, then Q α shoots a club through the approachable ordinals in κ α+1.

50 Coloring number Compactness The shift graph Sh 2 (λ): the vertex set V = [λ] 2, if x < y < z, {x, y} is joined to {y, z}. Theorem. Chr(Sh 2 (λ)) κ iff λ 2 κ. Proof. If λ > 2 κ, f : [λ] 2 κ, then there are x < y < z, f (x, y) = f (y, z) by the Erdős-Rado theorem. Enumerate κ 2 = {r ξ : ξ < 2 κ }. Define F(ξ, η) = α, i where α is the first difference of r ξ and r η, and i = 0 if r ξ (α) = 0 < r η (α) = 1, i = 1, OW. If ξ < η < ζ, F(ξ, η) = F(η, ζ), then r ξ (α) < r η (α) < r ζ (α), imp.

51 Coloring number Compactness General shift graph Sh n (λ): vertex set V = [λ] n, if x 0 < x 1 < < x n, then x 0,..., x n 1 is joined with {x 1,..., x n }. Theorem. Chr(Sh n (λ)) κ iff λ exp n 1 (κ).

52 Obligatory graphs Theorem. (Erdős Hajnal) If Chr(X) > ℵ 0, then each finite bipartite graph occurs in X and each finite nonbipartite graph can be omitted in graphs of arbitrarily large chromatic number. What are the obligatory families of finite graphs? Theorem. (Erdős Hajnal Shelah, Thomassen) If Chr(X) > ℵ 0, then X contains for some n all circuits C 2n+1, C 2n+3,....

53 Obligatory graphs If Chr(X) > ℵ 0 let f X : ω ω be defined as f X (n) is the number of vertices of the least n-chromatic subgraph of X. Clearly, f X (n) n and so f X. Question. (Erdős Hajnal) Can f X tend to arbitrarily fast? Theorem. (Shelah) Consistently for every function f : ω ω there is a graph X with X = Chr(X) = ℵ 1 and f X (n) f (n) (n 3).

54 Taylor conjecture Taylor conjecture (Taylor, Erdős Hajnal Shelah) If X is a graph, Chr(X) > ℵ 0, then for every cardinal λ there is a graph Y with the same finite subgraphs as X and Chr(Y ) > λ.

55 Taylor conjecture Notice: if X is a graph, Y is a graph with the same finite subgraphs and Chr(Y ) > λ, then Y embeds into the ultraproduct of its finite subgraphs, which embeds into the ultrapower of X. There is, therefore, an ultrapower Z of X with Chr(Z) > λ.

56 Taylor conjecture Theorem. It is consistent that there is a graph X with X = Chr(X) = ℵ 1 and if Y is a graph all whose finite subgraphs occur in X, then Chr(Y ) ℵ 2.

57 Taylor conjecture Proof. P adds a Cohen real, a function f : ω ω undominated by any function in V. Q adds a graph X on ω 1, Chr(X) = ω 1 with f X f. P Q = ℵ 1. Ass. Y V P,Q is a graph with the same finite graphs as X. Y splits into P Q = ℵ 1 graphs in V: Z p = {e : p e Y }. If Z Y, then f Y f Z, if Z V, then f Z V, so Chr(Z) is finite. So Y is the union of ℵ 1 finite chromatic graphs, and so Chr(Y ) 2 ℵ 1 = ℵ 2.

58 Taylor conjecture Theorem. It is consistent that if Chr(X) ℵ 2, then there are arbitrarily large chromatic graphs with the same finite subgraphs.

59 Taylor conjecture Proof. Let κ be such that if Chr(X) κ, then there are arbitrarily large chromatic graphs with the same finite subgraphs. Force with P = Coll(ω, κ). If X is a graph in V P with Chr(X) ℵ 2, then X is the union of P = ω graphs Y V, one has Chr(Y ) κ. Then there are arbitrarily large chromatic graphs Z in V with the same finite subgraphs as Y. But if λ κ, and V = Chr(Z) = λ, then V P = Chr(Z) = λ.

60 Jumping chrom. number Theorem (Shelah, Rinot) (2 λ = λ +, λ ) There is a graph X with X = Chr(X) = λ +, Chr(Y ) ω for Y X, Y < λ +.

61 Jumping chrom. number Theorem. (Shelah) (GCH) If λ > cf(λ) = µ + is a singular cardinal, then there is a cardinal, cofinality, and GCH preserving forcing extension in which there is a graph X with X = λ, Chr(X) = µ + on λ, such that Chr(Y ) µ holds for every subgraph Y of X with Y < λ.

62 Jumping chrom. number Theorem. (Shelah) (V=L) If κ is regular, not weakly compact, ω θ < κ, λ > cf(λ) = κ, then there is a graph X on λ with Chr(X) = θ +, such that if Y is a subgraph of X with Y < λ then Chr(Y ) θ.

63 Jumping chrom. number Theorem. (Foreman Laver) Relative to the existence of a huge cardinal, it is consistent that if X = Chr(X) = ℵ 2, then X contains a subgraph Y with Y = Chr(Y ) = ℵ 1. Proof. (Foreman) There is a model of GCH in which there is an ℵ 1 -dense, ω 1 -complete ideal I on ω 2. Let {A ξ : ξ < ω 1 } be dense in P(ω 1 )/I. Assume that f α : α ω is a good coloring of X α (α < ω 2 ). For each β < ω 2 there are ξ < ω 1, i < ω, such that for almost all α A ξ, f α (β) = i. F(β) = ξ, i is a good coloring.

64 Jumping chrom. number Conjecture. It is consistent that each graph X with X = ℵ 2, Chr(X) ℵ 1 contains a subgraph Y with Y = Chr(Y ) = ℵ 1. Implies non-ch.

65 Jumping chrom. number Theorem. (Shelah) Modulo the consistency of a supercompact cardinal it is consistent that (GCH and) each graph X = ℵ ω+1, Chr(X) ℵ 1 contains a subgraph Y with Y < ℵ ω, Chr(Y ) = ℵ 1.

66 Chrom. nu. of subgraphs Galvin s question: does the chromatic number have the Darboux-property: if Chr(X) = λ and κ < λ, then there is a subgraph Y X with Chr(Y ) = κ? W. l. o. g. ω < κ.

67 Chrom. nu. of subgraphs Theorem. (Galvin) If 2 ℵ 0 = 2ℵ 1 < 2ℵ 2, then there is a graph X, Chr(X) > ℵ 1, but it has no induced subgraph Y with Chr(Y ) = ℵ 1. Proof. X = Sh 2 (2 ℵ 2 ).

68 Chrom. nu. of subgraphs Theorem. It is consistent that there is a graph X with X = Chr(X) = ℵ 2 such that there is no subgraph Y X with Chr(Y ) = ℵ 1. Proof. Finite support iteration of length ω 3. Q 0 adds a graph X on ω 2 with finite conditions. For 0 < α < ω 3 let Y α be a subgraph of X with Chr(Y α ) = ℵ 1. Q α forces a good coloring of Y α with elements of ω, with finite approximations.

69 Chrom. nu. of subgraphs If X is a graph, let I(X) = { Chr(Y ) : Y induced subgr. in X } { 0, 1,..., ℵ0 }. Then I(X) is closed, and if λ I(X) is singular, then λ I(X). Further, if A is a set consisting of uncountable cardinals, then there is a cardinal, cofinality preserving forcing that adds a graph X such that I(X) = A.

70 Chrom. nu. of subgraphs If X is a graph, set S(X) = { Chr(Y ) : Y X } { 0, 1,..., ℵ 0 }. Then, if λ S(X) is singular, then λ S(X) and if λ S(X) is singular, then λ S(X). But S(X) is not necessarily closed at regular cardinals: Theorem. If the existence of a measurable cardinal is consistent, then it is consistent that S(X) is not closed at a regular cardinal.

71 Product Chromatic number of graph products If (V,X), (W,Y) are graphs, define their product (V W,X Y ) as X Y = {{ x, x, y, y } : {x, y} X, {x, y } Y } Hajnal: Chr(X Y ) = min(chr(x), Chr(Y ))?

72 Product Theorem. (Hajnal) If Chr(X) < ω Chr(X Y ), then Chr(X Y ) = Chr(X). Proof. Ass. Chr(X) = k + 1, Chr(X Y ) k. f : V W k good coloring. U ultrafilter on W extending co-finite-chromatic sets. For v V there is unique g(v) < k s.t. {w : f (v, w) = g(v)} U. There are v, v, {v, v } X, g(v) = g(v ). {w W : f (v, w) = f (v, w) = g(v)} U so it contains an edge: {w, w } Y. Now { v, w, v, w } X Y and f (v, w) = f (v, w ).

73 Product Theorem. (Hajnal) If Chr(X), Chr(Y ) > κ, Chr(X Y ) < κ, then Chr(X ) < κ for X X, X = κ.

74 Product Theorem. (Hajnal) If κ ω, there are {X i : i < κ + } on 2 κ s.t. Chr(X i ) = κ +, Chr(X i X j ) = κ (i j).

75 Product Proof. Set V = {f : α κ, inj, α < κ + }, {f, g} X iff f g.

76 Product Claim 1. Chr(X) = κ +. Proof. Let F κ be a good coloring. Define f α (α < κ + ): f 0 = 0, f α = {f β : β < α} (α limit), f α+1 = f α { α, F(f α ) }. Induction gives f β f α (β < α) and f α V. Then {f α : α < κ + } is an injective κ + κ function, contradiction! Define = { f, g V V,Dom(f ) = Dom(g)}. Claim 2. Chr(X X (V V )) κ. Proof. Define for Dom(f ) = α, Dom(g) = β, α < β, F( f, g ) = g(α) and similarly for β < α.

77 Product If A κ +, set V A = {f V : Dom(f ) A}, X A = X (V A) (V A). I = {A κ + : Chr(X A) κ} Claim 2. I is a κ + -complete, normal ideal on κ +. Ulam matrix implies there are κ + disjoint sets in I +, which gives the κ + graphs with κ-chromatic product.

78 Product Theorem. (Rinot) (2 λ = λ +, λ ) There are graphs {X i : i < λ + } with X i = Chr(X i ) = λ +, Chr(X i X j ) = ω.

79 List chromatic number The list chromatic number List(X) of a graph (V,X) is the least cardinal µ such that: if F(v) is arbitrary with F(v) = µ (v V) then there is a good coloring f of X such that f (v) F(v) (v V). Lemma. For every graph X Chr(X) List(X) Col(X) holds.

80 List chromatic number Theorem. If X is a bipartite graph on the bipartition classes A,B, with A = κ, B = 2 κ, N(x) = κ (x B), then List(X) > κ. Proof. First, let {F(a) : a A} be disjoint sets of size κ. For each choice function g {F(a) : a A} select an element b g B and set F(b g ) = {g(a) : {a, b g } X }. F(b g ) = κ by condition. If f (x) F(x) (x A B), then let g = f A, now f (b g ) cannot be any element of F(b g ).

81 List chromatic number Theorem. Consistently, for graphs of size ℵ 1 List(X) = ℵ 1 Chr(X) = ℵ 1. Proof. MA ω1 proves this. Let X be a graph on ω 1, Chr(X) ω, we want to prove that List(X) ω. Let h : ω ω be a good coloring of X, and let F(x) be given for each x ω 1, F(x) = ω. Set p P if p is a function, Dom(p) [ω 1 ] <ω, p(x) F(x) (x Dom(p)), and h(x) h(y) implies p(x) p(y). p p iff p p. The hard part is to show that (P, ) is ccc.

82 List chromatic number Theorem. Consistently, GCH holds and List(X) = Col(X) whenever the latter is infinite.

83 List chromatic number Theorem. (GCH) If List(X) is infinite, then Col(X) List(X) +. Theorem. (Kojman) Col(X) exp ω (List(X)) +.

84 Ramsey-theory Topological Ramsey-theory Definition. A topological K κ is a set of κ vetices plus a collection of paths between any two, disjoint except at ends. Theorem. (E H, 1964) If κ is infinite, n finite, then κ (TopK κ ) 2 n.

85 Ramsey-theory Topological The proof Assume X is the complete graph on V, κ = V. Let U be an ultrafilter on V such that if W V, W < V, then V W U. Let f : X {0, 1,..., n 1} be a coloring of X. Each v V has a principal color i(v) s. t. A(v) = {w : f (v, w) = i(v)} U. Principal color: B = {v : i(v) = i} U.

86 Ramsey-theory Topological Select the distinct vertices {v(α) : α < κ} and u(α, β) such that (1) i(v(α)) B, (2) u(α, β) A(v(α)) A(v(β)) (β < α). A topological K κ is given by {v(α) : α < κ} and the paths {v(α), u(α, β), v(β)}.

87 Ramsey-theory Topological Question (E H) κ (κ, TopK κ ) 2? Stronger than κ (TopK κ ) 2 n (n finite)

88 Ramsey-theory Topological Theorem. κ (κ, TopK κ ) 2 if and only if κ is regular and there is no κ-suslin tree.

89 Ramsey-theory Topological If κ is singular, let X be the disjoint union of cf(κ) complete graphs. In X, there is no independent set of size κ (or even cf(κ) + ), neither any connected subgraph of size κ, in particular, no topological K κ.

90 Ramsey-theory Topological If (T, ) is a tree (or any partially ordered set) then the comparison graph of (T, ) is the graph (T, X) where {t, t } X iff t < t or t < t.

91 Ramsey-theory Topological The comparison graph of a κ-suslin tree does not contain an independent set of size κ it would be an antichain of size κ.

92 Ramsey-theory Topological Assume that {v(α) : α < κ} gives a topological K κ with the connectings paths {p(α, β) : α < β < κ}. For any α < κ, there are only < κ nodes α < β < κ such that p(α, β) {v(β)} has any elements < v(α) (as there are < κ points < κ). There is a subsequence {α ξ : ξ < κ} such that if ξ < η, then all elements of p(α ξ, α η ) v(α η ) are above v(α ξ ) and so {v(α ξ ) : ξ < κ} is a κ-branch.

93 Ramsey-theory Balanced Can we have κ (TopK κ ) 2 ℵ 0? Not for κ = ℵ 1 as [ω 1 ] 2 is the union of countably many (graph theoretical) trees (Erdős Kakutani). So set κ = ℵ 2.

94 Ramsey-theory Balanced Theorem. If the existence of a huge cardinal is consistent then ω 2 (TopK ω2 ) 2 ℵ 0 is consistent.

95 Ramsey-theory Balanced ( ) If f : [ω 2 ] 2 ω, then there are i < ω and A [ω 2 ] ℵ 2 such that if α < β are in A, then {β < γ : f (α, γ) = f (β, γ) = i} = ℵ 2.

96 Ramsey-theory Balanced ( ) implies ω 2 (TopK ℵ2 ) 2 ℵ 0 : given A as above, select a(ξ) A, b(ξ, η) (ξ < η < ω 2 ) such that a(ξ) < a(η) < b(ξ, η), f (a(ξ), b(ξ, η)) = f (a(η), b(ξ, η)) = i and all a s and b s are distinct.

97 Ramsey-theory Balanced Theorem. (Foreman) Relative to the existence of a huge cardinal, it is consistent that there exists an ω 1 -complete, uniform, ℵ 1 -dense ideal I on ω 2. Implies ( ) by the E H principal color argument.

98 Ramsey-theory Balanced Theorem. (Shelah K) It is consistent with GCH that [ω 2 ] 2 is the union of countably many Suslin-trees. That is, K ℵ2 is the union of countably many graphs, each the comparison graph of an ℵ 2 -Suslin tree.

99 Ramsey-theory Induced X = (Y ) e µ denotes that if the edges of the graph X are colored with µ colors, then there is an induced copy of Y, all whose edges get the same color. Two simple examples: 1. K (2κ ) + = (K κ +)e κ 2. K ℵ1,ℵ 2 = (C 4 ) e ℵ 0

100 Ramsey-theory Induced Theorem. (Hajnal K) It is consistent that there is a graph X on ω 1 such that Y (X) e 2 holds for every graph Y. Theorem. (Shelah) It is consistent that for every graph X and cardinal µ, there is a graph Y such that Y = (X) e µ.

101 Ramsey-theory Induced Theorem. (Hajnal) For every finite graph X and cardinal µ, there is a graph Y such that Y = (X) e µ. What if X is countable?

102 Ramsey-theory No large cliques Theorem. (Shelah) If µ is a cardinal, then there is a forcing extension in which there is a graph X with no K 4, such that if the edges of X are colored with µ colors, then there is a monocolored K 3. Erdős: Does this hold in ZFC? Implies the existence of a finite graph X, no K 4, when 2-coloring, there is a monochromatic triangle.