# Finite subgraphs of uncountable graphs

April 28, 2020

## Transcript

1. ### Finite subgraphs of uncountable graphs Chris Lambie-Hanson Department of Mathematics

and Applied Mathematics Virginia Commonwealth University CMU Mathematical Logic Seminar 28 April 2020

4. ### Basic deﬁnitions Deﬁnition • A graph is a pair G

= (Σ, E), where E ⊆ [Σ]2.
5. ### Basic deﬁnitions Deﬁnition • A graph is a pair G

= (Σ, E), where E ⊆ [Σ]2. • A proper coloring of G is a function c with domain Σ such that c(σ) = c(τ) for all {σ, τ} ∈ E.
6. ### Basic deﬁnitions Deﬁnition • A graph is a pair G

= (Σ, E), where E ⊆ [Σ]2. • A proper coloring of G is a function c with domain Σ such that c(σ) = c(τ) for all {σ, τ} ∈ E. • The chromatic number of G, denoted χ(G), is the least cardinal χ such that there is a proper coloring c : Σ → χ.
7. ### Basic deﬁnitions Deﬁnition • A graph is a pair G

= (Σ, E), where E ⊆ [Σ]2. • A proper coloring of G is a function c with domain Σ such that c(σ) = c(τ) for all {σ, τ} ∈ E. • The chromatic number of G, denoted χ(G), is the least cardinal χ such that there is a proper coloring c : Σ → χ. • Given a natural number ≥ 3, a cycle of length in G is an injective sequence σi | i < from Σ such that • {σi , σi+1} ∈ E for all i < − 1; • {σ −1 , σ0} ∈ E.
8. ### Basic deﬁnitions Deﬁnition • A graph is a pair G

= (Σ, E), where E ⊆ [Σ]2. • A proper coloring of G is a function c with domain Σ such that c(σ) = c(τ) for all {σ, τ} ∈ E. • The chromatic number of G, denoted χ(G), is the least cardinal χ such that there is a proper coloring c : Σ → χ. • Given a natural number ≥ 3, a cycle of length in G is an injective sequence σi | i < from Σ such that • {σi , σi+1} ∈ E for all i < − 1; • {σ −1 , σ0} ∈ E. Note that the following are equivalent for a graph G:
9. ### Basic deﬁnitions Deﬁnition • A graph is a pair G

= (Σ, E), where E ⊆ [Σ]2. • A proper coloring of G is a function c with domain Σ such that c(σ) = c(τ) for all {σ, τ} ∈ E. • The chromatic number of G, denoted χ(G), is the least cardinal χ such that there is a proper coloring c : Σ → χ. • Given a natural number ≥ 3, a cycle of length in G is an injective sequence σi | i < from Σ such that • {σi , σi+1} ∈ E for all i < − 1; • {σ −1 , σ0} ∈ E. Note that the following are equivalent for a graph G: • G has no cycles of odd length;
10. ### Basic deﬁnitions Deﬁnition • A graph is a pair G

= (Σ, E), where E ⊆ [Σ]2. • A proper coloring of G is a function c with domain Σ such that c(σ) = c(τ) for all {σ, τ} ∈ E. • The chromatic number of G, denoted χ(G), is the least cardinal χ such that there is a proper coloring c : Σ → χ. • Given a natural number ≥ 3, a cycle of length in G is an injective sequence σi | i < from Σ such that • {σi , σi+1} ∈ E for all i < − 1; • {σ −1 , σ0} ∈ E. Note that the following are equivalent for a graph G: • G has no cycles of odd length; • G is bipartite;
11. ### Basic deﬁnitions Deﬁnition • A graph is a pair G

= (Σ, E), where E ⊆ [Σ]2. • A proper coloring of G is a function c with domain Σ such that c(σ) = c(τ) for all {σ, τ} ∈ E. • The chromatic number of G, denoted χ(G), is the least cardinal χ such that there is a proper coloring c : Σ → χ. • Given a natural number ≥ 3, a cycle of length in G is an injective sequence σi | i < from Σ such that • {σi , σi+1} ∈ E for all i < − 1; • {σ −1 , σ0} ∈ E. Note that the following are equivalent for a graph G: • G has no cycles of odd length; • G is bipartite; • χ(G) ≤ 2.

13. ### De Bruijn-Erd˝ os Theorem Theorem (De Bruijn-Erd˝ os) Suppose that

G is a graph, k is a natural number, and χ(H) ≤ k for every ﬁnite subgraph H of G.
14. ### De Bruijn-Erd˝ os Theorem Theorem (De Bruijn-Erd˝ os) Suppose that

G is a graph, k is a natural number, and χ(H) ≤ k for every ﬁnite subgraph H of G. Then χ(G) ≤ k.
15. ### De Bruijn-Erd˝ os Theorem Theorem (De Bruijn-Erd˝ os) Suppose that

G is a graph, k is a natural number, and χ(H) ≤ k for every ﬁnite subgraph H of G. Then χ(G) ≤ k. As a result, if a graph G has inﬁnite chromatic number, there is a function fG : N → N deﬁned by letting fG (k) be the least number of vertices in a subgraph of G with chromatic number k.
16. ### De Bruijn-Erd˝ os Theorem Theorem (De Bruijn-Erd˝ os) Suppose that

G is a graph, k is a natural number, and χ(H) ≤ k for every ﬁnite subgraph H of G. Then χ(G) ≤ k. As a result, if a graph G has inﬁnite chromatic number, there is a function fG : N → N deﬁned by letting fG (k) be the least number of vertices in a subgraph of G with chromatic number k. fG is clearly an increasing function.
17. ### De Bruijn-Erd˝ os Theorem Theorem (De Bruijn-Erd˝ os) Suppose that

G is a graph, k is a natural number, and χ(H) ≤ k for every ﬁnite subgraph H of G. Then χ(G) ≤ k. As a result, if a graph G has inﬁnite chromatic number, there is a function fG : N → N deﬁned by letting fG (k) be the least number of vertices in a subgraph of G with chromatic number k. fG is clearly an increasing function. Question How fast can fG grow for graphs G with large chromatic number?

19. ### Previous work Theorem (Erd˝ os, 1960s) For every function f

: N → N, there is a graph G such that |G| = χ(G) = ℵ0 and fG grows faster than f .
20. ### Previous work Theorem (Erd˝ os, 1960s) For every function f

: N → N, there is a graph G such that |G| = χ(G) = ℵ0 and fG grows faster than f . Theorem (Erd˝ os-Hajnal-Szemer´ edi, 1982) For every n ∈ N and every cardinal κ, there is a graph G such that χ(G) ≥ κ and fG grows faster than expn .
21. ### Previous work Theorem (Erd˝ os, 1960s) For every function f

: N → N, there is a graph G such that |G| = χ(G) = ℵ0 and fG grows faster than f . Theorem (Erd˝ os-Hajnal-Szemer´ edi, 1982) For every n ∈ N and every cardinal κ, there is a graph G such that χ(G) ≥ κ and fG grows faster than expn . Question (EHS, 1982) Is it true that, for every f : N → N, there is a graph G such that χ(G) > ℵ0 and fG grows faster than f ?
22. ### Countable vs. uncountable chromatic numbers Theorem (Erd˝ os; Erd˝ os-Hajnal)

• For every ∈ N, there is a graph G such that χ(G) = ℵ0 and G has no cycles of length less than .
23. ### Countable vs. uncountable chromatic numbers Theorem (Erd˝ os; Erd˝ os-Hajnal)

• For every ∈ N, there is a graph G such that χ(G) = ℵ0 and G has no cycles of length less than . • If χ(G) > ℵ0, then G contains every ﬁnite bipartite graph as a subgraph. In particular, it has cycles of all even lengths.
24. ### Countable vs. uncountable chromatic numbers Theorem (Erd˝ os; Erd˝ os-Hajnal)

• For every ∈ N, there is a graph G such that χ(G) = ℵ0 and G has no cycles of length less than . • If χ(G) > ℵ0, then G contains every ﬁnite bipartite graph as a subgraph. In particular, it has cycles of all even lengths. Theorem (R¨ odl; Komj´ ath-Shelah) • If χ(G) = ℵ0, then there is a triangle-free subgraph H of G such that χ(H) = ℵ0.
25. ### Countable vs. uncountable chromatic numbers Theorem (Erd˝ os; Erd˝ os-Hajnal)

• For every ∈ N, there is a graph G such that χ(G) = ℵ0 and G has no cycles of length less than . • If χ(G) > ℵ0, then G contains every ﬁnite bipartite graph as a subgraph. In particular, it has cycles of all even lengths. Theorem (R¨ odl; Komj´ ath-Shelah) • If χ(G) = ℵ0, then there is a triangle-free subgraph H of G such that χ(H) = ℵ0. • There is consistently a graph G such that χ(G) = ℵ1 but χ(H) ≤ ℵ0 for every triangle-free subgraph H of G.

27. ### Current status Theorem (Komj´ ath-Shelah, 2005) Consistently, for every f

: N → N, there is a graph G such that |G| = χ(G) = ℵ1 and fG grows faster than f .
28. ### Current status Theorem (Komj´ ath-Shelah, 2005) Consistently, for every f

: N → N, there is a graph G such that |G| = χ(G) = ℵ1 and fG grows faster than f . They force over a model of ♦ with a length-ω1 ﬁnite-support iteration of c.c.c. posets, dealing one at a time with each function f : N → N. CH holds in the resulting model.
29. ### Current status Theorem (Komj´ ath-Shelah, 2005) Consistently, for every f

: N → N, there is a graph G such that |G| = χ(G) = ℵ1 and fG grows faster than f . They force over a model of ♦ with a length-ω1 ﬁnite-support iteration of c.c.c. posets, dealing one at a time with each function f : N → N. CH holds in the resulting model. Main Theorem (CLH, 2019) For every f : N → N, there is a graph G such that |G| = 2ℵ0 , χ(G) = ℵ1, and fG (k) ≥ f (k) for all k ≥ 3.

31. ### Disjoint types Deﬁnition Let k ∈ N. • A disjoint

type of width n is a function t : 2n → 2 such that |t−1(0)| = |t−1(1)| = n.
32. ### Disjoint types Deﬁnition Let k ∈ N. • A disjoint

type of width n is a function t : 2n → 2 such that |t−1(0)| = |t−1(1)| = n. • If a, b ∈ [On]n are disjoint, then we say tp(a, b) = t if, letting a ∪ b be enumerated in increasing order as αi | i < 2n , we have a = {αi | t(i) = 0} and b = {αi | t(i) = 1}.
33. ### Disjoint types Deﬁnition Let k ∈ N. • A disjoint

type of width n is a function t : 2n → 2 such that |t−1(0)| = |t−1(1)| = n. • If a, b ∈ [On]n are disjoint, then we say tp(a, b) = t if, letting a ∪ b be enumerated in increasing order as αi | i < 2n , we have a = {αi | t(i) = 0} and b = {αi | t(i) = 1}.
34. ### tn s Deﬁnition Suppose that 1 ≤ s < n

are natural numbers. Then tn s is the disjoint type of width n consisting of s copies of 0, followed by n − s copies of 01, followed by s copies of 1.
35. ### tn s Deﬁnition Suppose that 1 ≤ s < n

are natural numbers. Then tn s is the disjoint type of width n consisting of s copies of 0, followed by n − s copies of 01, followed by s copies of 1.
36. ### Type graphs Deﬁnition Suppose that n is natural number, t

is a disjoint type of width n, and α is an ordinal.
37. ### Type graphs Deﬁnition Suppose that n is natural number, t

is a disjoint type of width n, and α is an ordinal. Then G(α, t) is the graph with vertex set [α]n and edge set E(α, t), where {a, b} ∈ E(α, t) if tp(a, b) = t or tp(b, a) = t.
38. ### Type graphs Deﬁnition Suppose that n is natural number, t

is a disjoint type of width n, and α is an ordinal. Then G(α, t) is the graph with vertex set [α]n and edge set E(α, t), where {a, b} ∈ E(α, t) if tp(a, b) = t or tp(b, a) = t. Theorem (Erd˝ os-Hajnal) Suppose that 1 ≤ s < n < ω and α is an inﬁnite ordinal.
39. ### Type graphs Deﬁnition Suppose that n is natural number, t

is a disjoint type of width n, and α is an ordinal. Then G(α, t) is the graph with vertex set [α]n and edge set E(α, t), where {a, b} ∈ E(α, t) if tp(a, b) = t or tp(b, a) = t. Theorem (Erd˝ os-Hajnal) Suppose that 1 ≤ s < n < ω and α is an inﬁnite ordinal. 1 χ(G(α, tn s )) = |α|.
40. ### Type graphs Deﬁnition Suppose that n is natural number, t

is a disjoint type of width n, and α is an ordinal. Then G(α, t) is the graph with vertex set [α]n and edge set E(α, t), where {a, b} ∈ E(α, t) if tp(a, b) = t or tp(b, a) = t. Theorem (Erd˝ os-Hajnal) Suppose that 1 ≤ s < n < ω and α is an inﬁnite ordinal. 1 χ(G(α, tn s )) = |α|. 2 If n ≥ 2s2 + 1, then G(α, tn s ) contains no odd cycles of length 2s + 1 or shorter.

46. ### Two useful facts Fact Suppose that G = (Σ, E)

is a graph and E = i∈I Ei . For each i ∈ I, let Gi = (Σ, Ei ). Then χ(G) ≤ i∈I χ(Gi ).
47. ### Two useful facts Fact Suppose that G = (Σ, E)

is a graph and E = i∈I Ei . For each i ∈ I, let Gi = (Σ, Ei ). Then χ(G) ≤ i∈I χ(Gi ). Fact Suppose that G and H are graphs and there is a graph homomorphism from G to H. If is a natural number and H contains no odd cycles of length or shorter, then G also contains no odd cycles of length or shorter.

49. ### The theorem Main Theorem For every function f : N

→ N, there is a graph G such that χ(G) = ℵ1 and fG (k) ≥ f (k) for all k ≥ 3.
50. ### The theorem Main Theorem For every function f : N

→ N, there is a graph G such that χ(G) = ℵ1 and fG (k) ≥ f (k) for all k ≥ 3. Proof: Fix f : N → N. We will actually construct G such that, for every k ∈ N and every subgraph H with at most f (k) vertices, we have χ(H) ≤ 2k+1.
51. ### The theorem Main Theorem For every function f : N

→ N, there is a graph G such that χ(G) = ℵ1 and fG (k) ≥ f (k) for all k ≥ 3. Proof: Fix f : N → N. We will actually construct G such that, for every k ∈ N and every subgraph H with at most f (k) vertices, we have χ(H) ≤ 2k+1. This suﬃces to prove the theorem.
52. ### The theorem Main Theorem For every function f : N

→ N, there is a graph G such that χ(G) = ℵ1 and fG (k) ≥ f (k) for all k ≥ 3. Proof: Fix f : N → N. We will actually construct G such that, for every k ∈ N and every subgraph H with at most f (k) vertices, we have χ(H) ≤ 2k+1. This suﬃces to prove the theorem. We construct a graph G = (Σ, E). Let us ﬁrst describe the set of vertices.
53. ### The vertices Σ is the set of all triples σ

= Mσ, fσ, Cσ such that
54. ### The vertices Σ is the set of all triples σ

= Mσ, fσ, Cσ such that 1 Mσ is a transitive element of H(ℵ1) such that δσ := Mσ ∩ ω1 is a limit ordinal;
55. ### The vertices Σ is the set of all triples σ

= Mσ, fσ, Cσ such that 1 Mσ is a transitive element of H(ℵ1) such that δσ := Mσ ∩ ω1 is a limit ordinal; 2 fσ : Mσ ∩ Σ → ω;
56. ### The vertices Σ is the set of all triples σ

= Mσ, fσ, Cσ such that 1 Mσ is a transitive element of H(ℵ1) such that δσ := Mσ ∩ ω1 is a limit ordinal; 2 fσ : Mσ ∩ Σ → ω; 3 Cσ is an ω-sequence of ordinals coﬁnal in δσ.
57. ### The vertices Σ is the set of all triples σ

= Mσ, fσ, Cσ such that 1 Mσ is a transitive element of H(ℵ1) such that δσ := Mσ ∩ ω1 is a limit ordinal; 2 fσ : Mσ ∩ Σ → ω; 3 Cσ is an ω-sequence of ordinals coﬁnal in δσ. • The M’s provide a partial ordering of Σ: σ ≺ τ iﬀ σ ∈ Mτ .
58. ### The vertices Σ is the set of all triples σ

= Mσ, fσ, Cσ such that 1 Mσ is a transitive element of H(ℵ1) such that δσ := Mσ ∩ ω1 is a limit ordinal; 2 fσ : Mσ ∩ Σ → ω; 3 Cσ is an ω-sequence of ordinals coﬁnal in δσ. • The M’s provide a partial ordering of Σ: σ ≺ τ iﬀ σ ∈ Mτ . • The f ’s will be used to anticipate potential proper colorings of G using only countably many colors in order to sabotage them.
59. ### The vertices Σ is the set of all triples σ

= Mσ, fσ, Cσ such that 1 Mσ is a transitive element of H(ℵ1) such that δσ := Mσ ∩ ω1 is a limit ordinal; 2 fσ : Mσ ∩ Σ → ω; 3 Cσ is an ω-sequence of ordinals coﬁnal in δσ. • The M’s provide a partial ordering of Σ: σ ≺ τ iﬀ σ ∈ Mτ . • The f ’s will be used to anticipate potential proper colorings of G using only countably many colors in order to sabotage them. • The C’s will place constraints on whether an edge can be drawn between two vertices, ensuring that the chromatic numbers of ﬁnite subgraphs do not grow too quickly.
60. ### Preliminary setup For each k ∈ N, let sk be

the least natural number s ≥ 1 such that 2s + 1 ≥ f (k), and let nk = 2s2 k + 1.
61. ### Preliminary setup For each k ∈ N, let sk be

the least natural number s ≥ 1 such that 2s + 1 ≥ f (k), and let nk = 2s2 k + 1. (Point: G(ω1, tnk sk ) has no odd cycles of length f (k) or shorter.)
62. ### Preliminary setup For each k ∈ N, let sk be

the least natural number s ≥ 1 such that 2s + 1 ≥ f (k), and let nk = 2s2 k + 1. (Point: G(ω1, tnk sk ) has no odd cycles of length f (k) or shorter.) Partition N into adjacent intervals I0, I1, I2, . . . such that |Ik | = nk
63. ### Preliminary setup For each k ∈ N, let sk be

the least natural number s ≥ 1 such that 2s + 1 ≥ f (k), and let nk = 2s2 k + 1. (Point: G(ω1, tnk sk ) has no odd cycles of length f (k) or shorter.) Partition N into adjacent intervals I0, I1, I2, . . . such that |Ik | = nk , i.e., I0 = {0, . . . , n0 − 1}, I1 = {n0, . . . , n0 + n1 − 1}, . . ..
64. ### Preliminary setup For each k ∈ N, let sk be

the least natural number s ≥ 1 such that 2s + 1 ≥ f (k), and let nk = 2s2 k + 1. (Point: G(ω1, tnk sk ) has no odd cycles of length f (k) or shorter.) Partition N into adjacent intervals I0, I1, I2, . . . such that |Ik | = nk , i.e., I0 = {0, . . . , n0 − 1}, I1 = {n0, . . . , n0 + n1 − 1}, . . .. Given an ω-sequence of ordinals C and a k ∈ N, we let C[Ik ] denote {α ∈ C | |C ∩ α| ∈ Ik }.

constraints.
66. ### The edges When constructing E, we will observe the following

constraints. • If {σ, τ} ∈ E, then σ ≺ τ or τ ≺ σ.
67. ### The edges When constructing E, we will observe the following

constraints. • If {σ, τ} ∈ E, then σ ≺ τ or τ ≺ σ. It thus suﬃces to deﬁne N≺ G (τ) := {σ ∈ Mτ | {σ, τ} ∈ E} for each τ ∈ Σ.
68. ### The edges When constructing E, we will observe the following

constraints. • If {σ, τ} ∈ E, then σ ≺ τ or τ ≺ σ. It thus suﬃces to deﬁne N≺ G (τ) := {σ ∈ Mτ | {σ, τ} ∈ E} for each τ ∈ Σ. • For all τ ∈ Σ and k ∈ N, there is at most one σ ∈ N≺ G (τ) such that fτ (σ) = k.
69. ### The edges When constructing E, we will observe the following

constraints. • If {σ, τ} ∈ E, then σ ≺ τ or τ ≺ σ. It thus suﬃces to deﬁne N≺ G (τ) := {σ ∈ Mτ | {σ, τ} ∈ E} for each τ ∈ Σ. • For all τ ∈ Σ and k ∈ N, there is at most one σ ∈ N≺ G (τ) such that fτ (σ) = k. • For all τ ∈ Σ and σ ∈ N≺ G (τ), if fτ (σ) = k, then, for all j ≤ k, we have tp(Cσ[Ij ], Cτ [Ij ]) = tnj sj .
70. ### The edges When constructing E, we will observe the following

constraints. • If {σ, τ} ∈ E, then σ ≺ τ or τ ≺ σ. It thus suﬃces to deﬁne N≺ G (τ) := {σ ∈ Mτ | {σ, τ} ∈ E} for each τ ∈ Σ. • For all τ ∈ Σ and k ∈ N, there is at most one σ ∈ N≺ G (τ) such that fτ (σ) = k. • For all τ ∈ Σ and σ ∈ N≺ G (τ), if fτ (σ) = k, then, for all j ≤ k, we have tp(Cσ[Ij ], Cτ [Ij ]) = tnj sj . Constructing E: For each τ ∈ Σ and each k ∈ N, ask whether there is σ ∈ Mτ ∩ Σ such that fτ (σ) = k and tp(Cσ[Ij ], Cτ [Ij ]) = tnj sj for all j ≤ k.
71. ### The edges When constructing E, we will observe the following

constraints. • If {σ, τ} ∈ E, then σ ≺ τ or τ ≺ σ. It thus suﬃces to deﬁne N≺ G (τ) := {σ ∈ Mτ | {σ, τ} ∈ E} for each τ ∈ Σ. • For all τ ∈ Σ and k ∈ N, there is at most one σ ∈ N≺ G (τ) such that fτ (σ) = k. • For all τ ∈ Σ and σ ∈ N≺ G (τ), if fτ (σ) = k, then, for all j ≤ k, we have tp(Cσ[Ij ], Cτ [Ij ]) = tnj sj . Constructing E: For each τ ∈ Σ and each k ∈ N, ask whether there is σ ∈ Mτ ∩ Σ such that fτ (σ) = k and tp(Cσ[Ij ], Cτ [Ij ]) = tnj sj for all j ≤ k. If the answer is “yes”, then choose one such σ and put {σ, τ} in E.
72. ### The edges When constructing E, we will observe the following

constraints. • If {σ, τ} ∈ E, then σ ≺ τ or τ ≺ σ. It thus suﬃces to deﬁne N≺ G (τ) := {σ ∈ Mτ | {σ, τ} ∈ E} for each τ ∈ Σ. • For all τ ∈ Σ and k ∈ N, there is at most one σ ∈ N≺ G (τ) such that fτ (σ) = k. • For all τ ∈ Σ and σ ∈ N≺ G (τ), if fτ (σ) = k, then, for all j ≤ k, we have tp(Cσ[Ij ], Cτ [Ij ]) = tnj sj . Constructing E: For each τ ∈ Σ and each k ∈ N, ask whether there is σ ∈ Mτ ∩ Σ such that fτ (σ) = k and tp(Cσ[Ij ], Cτ [Ij ]) = tnj sj for all j ≤ k. If the answer is “yes”, then choose one such σ and put {σ, τ} in E. If the answer is “no”, then do nothing for this pair (τ, k).
73. ### An edge decomposition Given k ∈ N, let Ek (resp.

E≥k ) denote the set of edges {σ, τ} ∈ E such that σ ≺ τ and fτ (σ) = k (resp. fτ (σ) ≥ k).
74. ### An edge decomposition Given k ∈ N, let Ek (resp.

E≥k ) denote the set of edges {σ, τ} ∈ E such that σ ≺ τ and fτ (σ) = k (resp. fτ (σ) ≥ k). Let Gk = (Σ, Ek ) and G≥k = (Σ, E≥k ) (and similarly deﬁne Hk and H≥k for subgraphs H of G).
75. ### An edge decomposition Given k ∈ N, let Ek (resp.

E≥k ) denote the set of edges {σ, τ} ∈ E such that σ ≺ τ and fτ (σ) = k (resp. fτ (σ) ≥ k). Let Gk = (Σ, Ek ) and G≥k = (Σ, E≥k ) (and similarly deﬁne Hk and H≥k for subgraphs H of G). Observe: • For each τ ∈ Σ, there is at most one σ ≺ τ such that {σ, τ} ∈ Ek .
76. ### An edge decomposition Given k ∈ N, let Ek (resp.

E≥k ) denote the set of edges {σ, τ} ∈ E such that σ ≺ τ and fτ (σ) = k (resp. fτ (σ) ≥ k). Let Gk = (Σ, Ek ) and G≥k = (Σ, E≥k ) (and similarly deﬁne Hk and H≥k for subgraphs H of G). Observe: • For each τ ∈ Σ, there is at most one σ ≺ τ such that {σ, τ} ∈ Ek . As a result, Gk has no cycles.
77. ### An edge decomposition Given k ∈ N, let Ek (resp.

E≥k ) denote the set of edges {σ, τ} ∈ E such that σ ≺ τ and fτ (σ) = k (resp. fτ (σ) ≥ k). Let Gk = (Σ, Ek ) and G≥k = (Σ, E≥k ) (and similarly deﬁne Hk and H≥k for subgraphs H of G). Observe: • For each τ ∈ Σ, there is at most one σ ≺ τ such that {σ, τ} ∈ Ek . As a result, Gk has no cycles. • The map σ → Cσ[Ik ] induces a graph homomorphism from G≥k to G(ω1, tnk sk ).
78. ### An edge decomposition Given k ∈ N, let Ek (resp.

E≥k ) denote the set of edges {σ, τ} ∈ E such that σ ≺ τ and fτ (σ) = k (resp. fτ (σ) ≥ k). Let Gk = (Σ, Ek ) and G≥k = (Σ, E≥k ) (and similarly deﬁne Hk and H≥k for subgraphs H of G). Observe: • For each τ ∈ Σ, there is at most one σ ≺ τ such that {σ, τ} ∈ Ek . As a result, Gk has no cycles. • The map σ → Cσ[Ik ] induces a graph homomorphism from G≥k to G(ω1, tnk sk ). As a result, G≥k has no odd cycles of length f (k) or shorter.
79. ### Finite subgraphs Fix k ∈ N and a subgraph H

of G with at most f (k) vertices.
80. ### Finite subgraphs Fix k ∈ N and a subgraph H

of G with at most f (k) vertices. We get an edge-decomposition of H as H = H≥k ∪ j<k Hj .
81. ### Finite subgraphs Fix k ∈ N and a subgraph H

of G with at most f (k) vertices. We get an edge-decomposition of H as H = H≥k ∪ j<k Hj . • For each j < k, Hj has no cycles, so χ(Hj ) ≤ 2.
82. ### Finite subgraphs Fix k ∈ N and a subgraph H

of G with at most f (k) vertices. We get an edge-decomposition of H as H = H≥k ∪ j<k Hj . • For each j < k, Hj has no cycles, so χ(Hj ) ≤ 2. • H≥k has no odd cycles of length f (k) or shorter.
83. ### Finite subgraphs Fix k ∈ N and a subgraph H

of G with at most f (k) vertices. We get an edge-decomposition of H as H = H≥k ∪ j<k Hj . • For each j < k, Hj has no cycles, so χ(Hj ) ≤ 2. • H≥k has no odd cycles of length f (k) or shorter. But H≥k has at most f (k) vertices, so H≥k has no odd cycles.
84. ### Finite subgraphs Fix k ∈ N and a subgraph H

of G with at most f (k) vertices. We get an edge-decomposition of H as H = H≥k ∪ j<k Hj . • For each j < k, Hj has no cycles, so χ(Hj ) ≤ 2. • H≥k has no odd cycles of length f (k) or shorter. But H≥k has at most f (k) vertices, so H≥k has no odd cycles. Thus, χ(H≥k ) ≤ 2.
85. ### Finite subgraphs Fix k ∈ N and a subgraph H

of G with at most f (k) vertices. We get an edge-decomposition of H as H = H≥k ∪ j<k Hj . • For each j < k, Hj has no cycles, so χ(Hj ) ≤ 2. • H≥k has no odd cycles of length f (k) or shorter. But H≥k has at most f (k) vertices, so H≥k has no odd cycles. Thus, χ(H≥k ) ≤ 2. But then χ(H) ≤ χ(H≥k ) · j<k χ(Hj ) ≤ 2k+1, as desired.

87. ### Chromatic number It remains to show that χ(G) = ℵ1.

χ(G) ≤ ℵ1 is easy.
88. ### Chromatic number It remains to show that χ(G) = ℵ1.

χ(G) ≤ ℵ1 is easy. Thus, suppose that f : Σ → N is given. We show that f is not a proper coloring.
89. ### Chromatic number It remains to show that χ(G) = ℵ1.

χ(G) ≤ ℵ1 is easy. Thus, suppose that f : Σ → N is given. We show that f is not a proper coloring. Let Ni | i ≤ ω be a continuous ∈-increasing chain of countable elementary submodels of (H(c+), ∈, , G, f ).
90. ### Chromatic number It remains to show that χ(G) = ℵ1.

χ(G) ≤ ℵ1 is easy. Thus, suppose that f : Σ → N is given. We show that f is not a proper coloring. Let Ni | i ≤ ω be a continuous ∈-increasing chain of countable elementary submodels of (H(c+), ∈, , G, f ). For i ≤ ω, let Mi = Ni ∩ H(ℵ1) and δi = Mi ∩ ω1.
91. ### Chromatic number It remains to show that χ(G) = ℵ1.

χ(G) ≤ ℵ1 is easy. Thus, suppose that f : Σ → N is given. We show that f is not a proper coloring. Let Ni | i ≤ ω be a continuous ∈-increasing chain of countable elementary submodels of (H(c+), ∈, , G, f ). For i ≤ ω, let Mi = Ni ∩ H(ℵ1) and δi = Mi ∩ ω1. Let C = {δi | i < ω} and τ = Mω, f (Mω ∩ Σ), C .
92. ### Chromatic number It remains to show that χ(G) = ℵ1.

χ(G) ≤ ℵ1 is easy. Thus, suppose that f : Σ → N is given. We show that f is not a proper coloring. Let Ni | i ≤ ω be a continuous ∈-increasing chain of countable elementary submodels of (H(c+), ∈, , G, f ). For i ≤ ω, let Mi = Ni ∩ H(ℵ1) and δi = Mi ∩ ω1. Let C = {δi | i < ω} and τ = Mω, f (Mω ∩ Σ), C . Then τ ∈ Σ. Let k = f (τ).
93. ### Multiple reﬂections Claim: For every n < ω and every

disjoint type t of width n, there is σ ∈ Mω such that f (σ) = k and tp(Cσ[n], C[n]) = t.
94. ### Multiple reﬂections Claim: For every n < ω and every

disjoint type t of width n, there is σ ∈ Mω such that f (σ) = k and tp(Cσ[n], C[n]) = t. Proof sketch: For concreteness, we prove the claim for t = t3 1 = 001011.
95. ### Multiple reﬂections Claim: For every n < ω and every

disjoint type t of width n, there is σ ∈ Mω such that f (σ) = k and tp(Cσ[n], C[n]) = t. Proof sketch: For concreteness, we prove the claim for t = t3 1 = 001011. Let ∃∞γ . . . abbreviate “there are unboundedly many γ < ω1 . . .”.
96. ### Multiple reﬂections Claim: For every n < ω and every

disjoint type t of width n, there is σ ∈ Mω such that f (σ) = k and tp(Cσ[n], C[n]) = t. Proof sketch: For concreteness, we prove the claim for t = t3 1 = 001011. Let ∃∞γ . . . abbreviate “there are unboundedly many γ < ω1 . . .”. For every β < δ2, the following sentence holds in H(c+): ∃γ2 > β ∃σ (f (σ) = k ∧ Cσ = {δ0, δ1, γ2}), as witnessed by δ2 and τ.
97. ### Multiple reﬂections Claim: For every n < ω and every

disjoint type t of width n, there is σ ∈ Mω such that f (σ) = k and tp(Cσ[n], C[n]) = t. Proof sketch: For concreteness, we prove the claim for t = t3 1 = 001011. Let ∃∞γ . . . abbreviate “there are unboundedly many γ < ω1 . . .”. For every β < δ2, the following sentence holds in H(c+): ∃γ2 > β ∃σ (f (σ) = k ∧ Cσ = {δ0, δ1, γ2}), as witnessed by δ2 and τ. By elementarity, the following holds in N2: ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {δ0, δ1, γ2}),
98. ### Multiple reﬂections Claim: For every n < ω and every

disjoint type t of width n, there is σ ∈ Mω such that f (σ) = k and tp(Cσ[n], C[n]) = t. Proof sketch: For concreteness, we prove the claim for t = t3 1 = 001011. Let ∃∞γ . . . abbreviate “there are unboundedly many γ < ω1 . . .”. For every β < δ2, the following sentence holds in H(c+): ∃γ2 > β ∃σ (f (σ) = k ∧ Cσ = {δ0, δ1, γ2}), as witnessed by δ2 and τ. By elementarity, the following holds in N2: ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {δ0, δ1, γ2}), and hence also in H(c+).
99. ### Moving down Moving one step down, for every β <

δ1, the following sentence holds in H(c+): ∃γ1 > β ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {δ0, γ1, γ2}), as witnessed by δ1.
100. ### Moving down Moving one step down, for every β <

δ1, the following sentence holds in H(c+): ∃γ1 > β ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {δ0, γ1, γ2}), as witnessed by δ1. By elementarity, the following holds in N1, and hence in H(c+): ∃∞γ1 ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {δ0, γ1, γ2}).
101. ### Moving down Moving one step down, for every β <

δ1, the following sentence holds in H(c+): ∃γ1 > β ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {δ0, γ1, γ2}), as witnessed by δ1. By elementarity, the following holds in N1, and hence in H(c+): ∃∞γ1 ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {δ0, γ1, γ2}). Similarly, in N0 and H(c+) we get: ∃∞γ0 ∃∞γ1 ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {γ0, γ1, γ2}).
102. ### Moving back up We have ∃∞γ0 ∃∞γ1 ∃∞γ2 ∃σ (f

(σ) = k ∧ Cσ = {γ0, γ1, γ2}). and t = t3 1 = 001011.
103. ### Moving back up We have ∃∞γ0 ∃∞γ1 ∃∞γ2 ∃σ (f

(σ) = k ∧ Cσ = {γ0, γ1, γ2}). and t = t3 1 = 001011. Working inside N0, choose γ∗ 0 < γ∗ 1 < δ0 such that ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {γ∗ 0 , γ∗ 1 , γ2}) holds (in N0 and hence also in N1).
104. ### Moving back up We have ∃∞γ0 ∃∞γ1 ∃∞γ2 ∃σ (f

(σ) = k ∧ Cσ = {γ0, γ1, γ2}). and t = t3 1 = 001011. Working inside N0, choose γ∗ 0 < γ∗ 1 < δ0 such that ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {γ∗ 0 , γ∗ 1 , γ2}) holds (in N0 and hence also in N1). Next, working in N1, choose γ∗ 2 and σ such that δ0 < γ∗ 2 < δ1, f (σ) = k, and Cσ = {γ∗ 0 , γ∗ 1 , γ∗ 2 }.
105. ### Moving back up We have ∃∞γ0 ∃∞γ1 ∃∞γ2 ∃σ (f

(σ) = k ∧ Cσ = {γ0, γ1, γ2}). and t = t3 1 = 001011. Working inside N0, choose γ∗ 0 < γ∗ 1 < δ0 such that ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ = {γ∗ 0 , γ∗ 1 , γ2}) holds (in N0 and hence also in N1). Next, working in N1, choose γ∗ 2 and σ such that δ0 < γ∗ 2 < δ1, f (σ) = k, and Cσ = {γ∗ 0 , γ∗ 1 , γ∗ 2 }. Then σ ∈ Mω, f (σ) = k, and tp(Cσ[n], C[n]) = t, as required. Claim
106. ### Finishing the proof Applying the Claim to the type t

= tn0 s0 tn1 s1 . . . tnk sk , there is σ ∈ Mω such that f (σ) = k and tp(Cσ[Ij ], C[Ij ]) = tnj sj for all j ≤ k.
107. ### Finishing the proof Applying the Claim to the type t

= tn0 s0 tn1 s1 . . . tnk sk , there is σ ∈ Mω such that f (σ) = k and tp(Cσ[Ij ], C[Ij ]) = tnj sj for all j ≤ k. Thus, when we were building G, we chose such a σ and added {σ, τ} to E. But then we have f (σ) = k = f (τ), so f is not a proper coloring. Theorem

109. ### On optimality Two natural questions arise regarding the optimality of

our main theorem.
110. ### On optimality Two natural questions arise regarding the optimality of

our main theorem. Question Is it true that for every f : N → N, there is a graph G such that |G| = χ(G) = ℵ1 and fG grows faster than f ?
111. ### On optimality Two natural questions arise regarding the optimality of

our main theorem. Question Is it true that for every f : N → N, there is a graph G such that |G| = χ(G) = ℵ1 and fG grows faster than f ? Question Is it true that for ever f : N → N and every cardinal κ, there is a graph G such that χ(G) ≥ κ and fG grows faster than f ?
112. ### On optimality Two natural questions arise regarding the optimality of

our main theorem. Question Is it true that for every f : N → N, there is a graph G such that |G| = χ(G) = ℵ1 and fG grows faster than f ? Question Is it true that for ever f : N → N and every cardinal κ, there is a graph G such that χ(G) ≥ κ and fG grows faster than f ? What about just κ = ℵ2?
113. ### Other questions We end with some other questions of Erd˝

os and Hajnal about ﬁnite subgraphs of uncountably chromatic graphs.
114. ### Other questions We end with some other questions of Erd˝

os and Hajnal about ﬁnite subgraphs of uncountably chromatic graphs. Question Suppose that G and H are uncountably chromatic graphs. Must they have a common 4-chromatic subgraph?
115. ### Other questions We end with some other questions of Erd˝

os and Hajnal about ﬁnite subgraphs of uncountably chromatic graphs. Question Suppose that G and H are uncountably chromatic graphs. Must they have a common 4-chromatic subgraph? Question Is there an uncountably chromatic graph G such that every triangle-free subgraph of G has countable chromatic number?
116. ### Other questions We end with some other questions of Erd˝

os and Hajnal about ﬁnite subgraphs of uncountably chromatic graphs. Question Suppose that G and H are uncountably chromatic graphs. Must they have a common 4-chromatic subgraph? Question Is there an uncountably chromatic graph G such that every triangle-free subgraph of G has countable chromatic number? Question Is there a K4-free graph that cannot be written as the union of countably many triangle-free graphs?
117. ### References Erd˝ os, Paul. Some of my favourite unsolved problems.

A tribute to Paul Erd˝ os, 467–478, Cambridge Univ. Press, Cambridge, 1990.
118. ### References Erd˝ os, Paul. Some of my favourite unsolved problems.

A tribute to Paul Erd˝ os, 467–478, Cambridge Univ. Press, Cambridge, 1990. Lambie-Hanson, Chris. On the growth rate of chromatic numbers of ﬁnite subgraphs. Adv. Math. To appear, 2020. https://arxiv.org/pdf/1902.08177.pdf
119. ### References Erd˝ os, Paul. Some of my favourite unsolved problems.

A tribute to Paul Erd˝ os, 467–478, Cambridge Univ. Press, Cambridge, 1990. Lambie-Hanson, Chris. On the growth rate of chromatic numbers of ﬁnite subgraphs. Adv. Math. To appear, 2020. https://arxiv.org/pdf/1902.08177.pdf All artwork on these slides by Joan Mir´ o.