Finite subgraphs of uncountable graphs

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
2. I. Graph theoretic background
3. Basic definitions
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.
12. De Bruijn-Erd˝ os Theorem
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?
18. Previous work
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.
26. Current status
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.
30. II. Type graphs
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.
41. A special case G(α, t3 1 ) is triangle-free.
42. A special case G(α, t3 1 ) is triangle-free.
43. A special case G(α, t3 1 ) is triangle-free.
44. A special case G(α, t3 1 ) is triangle-free.
45. A special case G(α, t3 1 ) is triangle-free.
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.
48. III. The main construction
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 }.
65. The edges When constructing E, we will observe the following 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.
86. Chromatic number It remains to show that χ(G) = ℵ1.
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[3] = {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[3] = {δ0, δ1, γ2}), as witnessed by δ2 and τ. By elementarity, the following holds in N2: ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ[3] = {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[3] = {δ0, δ1, γ2}), as witnessed by δ2 and τ. By elementarity, the following holds in N2: ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ[3] = {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[3] = {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[3] = {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[3] = {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[3] = {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[3] = {δ0, γ1, γ2}). Similarly, in N0 and H(c+) we get: ∃∞γ0 ∃∞γ1 ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ[3] = {γ0, γ1, γ2}).
102. Moving back up We have ∃∞γ0 ∃∞γ1 ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ[3] = {γ0, γ1, γ2}). and t = t3 1 = 001011.
103. Moving back up We have ∃∞γ0 ∃∞γ1 ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ[3] = {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[3] = {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[3] = {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[3] = {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[3] = {γ∗ 0 , γ∗ 1 , γ∗ 2 }.
105. Moving back up We have ∃∞γ0 ∃∞γ1 ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ[3] = {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[3] = {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[3] = {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
108. IV. Further questions
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.
120. Thank you!