Reflections on Graph Coloring

Transcript

1. Reflections on graph coloring (Joint work with Assaf Rinot) Chris Lambie-Hanson Department of Mathematics Bar-Ilan University MAMLS Logic Friday CUNY Graduate Center 27 October 2017
2. ‘The purest and most thoughtful minds are those which love colour the most.’ -John Ruskin
3. I. Chromatic and coloring numbers
4. Graphs Definition 1 A graph is a pair G = (V, E) such that E ⊆ [V]2.
5. Graphs Definition 1 A graph is a pair G = (V, E) such that E ⊆ [V]2. 2 If G = (V, E) is a graph and u ∈ V, then NG(u) = {v ∈ V | {u, v} ∈ E}.
6. Graphs Definition 1 A graph is a pair G = (V, E) such that E ⊆ [V]2. 2 If G = (V, E) is a graph and u ∈ V, then NG(u) = {v ∈ V | {u, v} ∈ E}.
7. Chromatic number Definition Let G = (V, E) be a graph. 1 A chromatic coloring of G is a function c, with domain V, such that, for all {u, v} ∈ E, we have c(u) = c(v).
8. Chromatic number Definition Let G = (V, E) be a graph. 1 A chromatic coloring of G is a function c, with domain V, such that, for all {u, v} ∈ E, we have c(u) = c(v). 2 The chromatic number of G (Chr(G)) is the least cardinal χ such that there is a chromatic coloring c : V → χ.
9. Chromatic number Definition Let G = (V, E) be a graph. 1 A chromatic coloring of G is a function c, with domain V, such that, for all {u, v} ∈ E, we have c(u) = c(v). 2 The chromatic number of G (Chr(G)) is the least cardinal χ such that there is a chromatic coloring c : V → χ.
10. Coloring number Definition Let G = (V, E) be a graph. 1 If is a well-ordering of V and u ∈ V, then N G (u) = {v u | {u, v} ∈ E}.
11. Coloring number Definition Let G = (V, E) be a graph. 1 If is a well-ordering of V and u ∈ V, then N G (u) = {v u | {u, v} ∈ E}. 2 The coloring number of G (Col(G)) is the least cardinal χ such that there is a well-ordering of V such that, for all u ∈ V, we have |N G (u)| < χ.
12. Coloring number Definition Let G = (V, E) be a graph. 1 If is a well-ordering of V and u ∈ V, then N G (u) = {v u | {u, v} ∈ E}. 2 The coloring number of G (Col(G)) is the least cardinal χ such that there is a well-ordering of V such that, for all u ∈ V, we have |N G (u)| < χ.
13. Coloring number Definition Let G = (V, E) be a graph. 1 If is a well-ordering of V and u ∈ V, then N G (u) = {v u | {u, v} ∈ E}. 2 The coloring number of G (Col(G)) is the least cardinal χ such that there is a well-ordering of V such that, for all u ∈ V, we have |N G (u)| < χ. We easily have Chr(G) ≤ Col(G).
14. Coloring number Definition Let G = (V, E) be a graph. 1 If is a well-ordering of V and u ∈ V, then N G (u) = {v u | {u, v} ∈ E}. 2 The coloring number of G (Col(G)) is the least cardinal χ such that there is a well-ordering of V such that, for all u ∈ V, we have |N G (u)| < χ. We easily have Chr(G) ≤ Col(G).
15. Coloring number Definition Let G = (V, E) be a graph. 1 If is a well-ordering of V and u ∈ V, then N G (u) = {v u | {u, v} ∈ E}. 2 The coloring number of G (Col(G)) is the least cardinal χ such that there is a well-ordering of V such that, for all u ∈ V, we have |N G (u)| < χ. We easily have Chr(G) ≤ Col(G).
16. Coloring number Definition Let G = (V, E) be a graph. 1 If is a well-ordering of V and u ∈ V, then N G (u) = {v u | {u, v} ∈ E}. 2 The coloring number of G (Col(G)) is the least cardinal χ such that there is a well-ordering of V such that, for all u ∈ V, we have |N G (u)| < χ. We easily have Chr(G) ≤ Col(G).
17. Coloring number Definition Let G = (V, E) be a graph. 1 If is a well-ordering of V and u ∈ V, then N G (u) = {v u | {u, v} ∈ E}. 2 The coloring number of G (Col(G)) is the least cardinal χ such that there is a well-ordering of V such that, for all u ∈ V, we have |N G (u)| < χ. We easily have Chr(G) ≤ Col(G).
18. Coloring number Definition Let G = (V, E) be a graph. 1 If is a well-ordering of V and u ∈ V, then N G (u) = {v u | {u, v} ∈ E}. 2 The coloring number of G (Col(G)) is the least cardinal χ such that there is a well-ordering of V such that, for all u ∈ V, we have |N G (u)| < χ. We easily have Chr(G) ≤ Col(G).
19. II. Reflection/Compactness
20. Reflection/compactness principles A reflection principle about a cardinal κ is an assertion of the form: Whenever A is a [structure] of size κ with [property], then A has a “small” substructure with [property].
21. Reflection/compactness principles A reflection principle about a cardinal κ is an assertion of the form: Whenever A is a [structure] of size κ with [property], then A has a “small” substructure with [property]. The dual notion is that of a compactness principle: Whenever A is a [structure] of size κ such that every “small” substructure has [property], then A has [property].
22. Reflection/compactness principles A reflection principle about a cardinal κ is an assertion of the form: Whenever A is a [structure] of size κ with [property], then A has a “small” substructure with [property]. The dual notion is that of a compactness principle: Whenever A is a [structure] of size κ such that every “small” substructure has [property], then A has [property]. Examples of reflection principles • (Stationary reflection) For every stationary S ⊆ κ, there is α ∈ κ ∩ cof(> ω) such that S ∩ α is stationary in α.
23. Reflection/compactness principles A reflection principle about a cardinal κ is an assertion of the form: Whenever A is a [structure] of size κ with [property], then A has a “small” substructure with [property]. The dual notion is that of a compactness principle: Whenever A is a [structure] of size κ such that every “small” substructure has [property], then A has [property]. Examples of reflection principles • (Stationary reflection) For every stationary S ⊆ κ, there is α ∈ κ ∩ cof(> ω) such that S ∩ α is stationary in α. • If G is an abelian group of size κ such that every subgroup of G of size < κ is free, then G is free.
24. Compactness heuristics • ℵ0 displays a high degree of compactness:
25. Compactness heuristics • ℵ0 displays a high degree of compactness: Compactness theorem for first-order logic
26. Compactness heuristics • ℵ0 displays a high degree of compactness: Compactness theorem for first-order logic K¨ onig’s infinity lemma
27. Compactness heuristics • ℵ0 displays a high degree of compactness: Compactness theorem for first-order logic K¨ onig’s infinity lemma Ramsey’s theorem
28. Compactness heuristics • ℵ0 displays a high degree of compactness: Compactness theorem for first-order logic K¨ onig’s infinity lemma Ramsey’s theorem • ℵ1 displays a high degree of incompactness:
29. Compactness heuristics • ℵ0 displays a high degree of compactness: Compactness theorem for first-order logic K¨ onig’s infinity lemma Ramsey’s theorem • ℵ1 displays a high degree of incompactness: Existence of ℵ1 -Aronszajn trees
30. Compactness heuristics • ℵ0 displays a high degree of compactness: Compactness theorem for first-order logic K¨ onig’s infinity lemma Ramsey’s theorem • ℵ1 displays a high degree of incompactness: Existence of ℵ1 -Aronszajn trees Existence of a J´ onsson group of size ℵ1
31. Compactness heuristics • ℵ0 displays a high degree of compactness: Compactness theorem for first-order logic K¨ onig’s infinity lemma Ramsey’s theorem • ℵ1 displays a high degree of incompactness: Existence of ℵ1 -Aronszajn trees Existence of a J´ onsson group of size ℵ1 • G¨ odel’s universe L exhibits a high degree of incompactness.
32. Compactness heuristics • ℵ0 displays a high degree of compactness: Compactness theorem for first-order logic K¨ onig’s infinity lemma Ramsey’s theorem • ℵ1 displays a high degree of incompactness: Existence of ℵ1 -Aronszajn trees Existence of a J´ onsson group of size ℵ1 • G¨ odel’s universe L exhibits a high degree of incompactness. • Large cardinals imply instances of compactness.
33. Compactness heuristics • ℵ0 displays a high degree of compactness: Compactness theorem for first-order logic K¨ onig’s infinity lemma Ramsey’s theorem • ℵ1 displays a high degree of incompactness: Existence of ℵ1 -Aronszajn trees Existence of a J´ onsson group of size ℵ1 • G¨ odel’s universe L exhibits a high degree of incompactness. • Large cardinals imply instances of compactness. • Many interesting questions revolve around the extent to which “small” uncountable cardinals can exhibit compactness.
34. De Bruijn-Erd˝ os theorem Theorem (de Bruijn-Erd˝ os) Suppose that G = (V, E) is a graph, k < ω, and every finite subgraph of G has chromatic number ≤ k. Then Chr(G) ≤ k.
35. De Bruijn-Erd˝ os theorem Theorem (de Bruijn-Erd˝ os) Suppose that G = (V, E) is a graph, k < ω, and every finite subgraph of G has chromatic number ≤ k. Then Chr(G) ≤ k. Proof. Let Pω(V) denote the set of finite subsets of V. For each x ∈ Pω(V), let Gx = (x, E ∩ [x]2) be the induced subgraph on x, and let cx : x → k witness that Chr(Gx ) ≤ k.
36. De Bruijn-Erd˝ os theorem Theorem (de Bruijn-Erd˝ os) Suppose that G = (V, E) is a graph, k < ω, and every finite subgraph of G has chromatic number ≤ k. Then Chr(G) ≤ k. Proof. Let Pω(V) denote the set of finite subsets of V. For each x ∈ Pω(V), let Gx = (x, E ∩ [x]2) be the induced subgraph on x, and let cx : x → k witness that Chr(Gx ) ≤ k. Let U be a fine ultrafilter over Pω(V), i.e., an ultrafilter such that, for all u ∈ V, Zu := {x ∈ Pω(V) | u ∈ x} ∈ U.
37. De Bruijn-Erd˝ os theorem Theorem (de Bruijn-Erd˝ os) Suppose that G = (V, E) is a graph, k < ω, and every finite subgraph of G has chromatic number ≤ k. Then Chr(G) ≤ k. Proof. Let Pω(V) denote the set of finite subsets of V. For each x ∈ Pω(V), let Gx = (x, E ∩ [x]2) be the induced subgraph on x, and let cx : x → k witness that Chr(Gx ) ≤ k. Let U be a fine ultrafilter over Pω(V), i.e., an ultrafilter such that, for all u ∈ V, Zu := {x ∈ Pω(V) | u ∈ x} ∈ U. For each u ∈ V, fix mu < k such that {y ∈ Zu | cy (u) = mu} ∈ U.
38. De Bruijn-Erd˝ os theorem Theorem (de Bruijn-Erd˝ os) Suppose that G = (V, E) is a graph, k < ω, and every finite subgraph of G has chromatic number ≤ k. Then Chr(G) ≤ k. Proof. Let Pω(V) denote the set of finite subsets of V. For each x ∈ Pω(V), let Gx = (x, E ∩ [x]2) be the induced subgraph on x, and let cx : x → k witness that Chr(Gx ) ≤ k. Let U be a fine ultrafilter over Pω(V), i.e., an ultrafilter such that, for all u ∈ V, Zu := {x ∈ Pω(V) | u ∈ x} ∈ U. For each u ∈ V, fix mu < k such that {y ∈ Zu | cy (u) = mu} ∈ U. Define c : V → k by letting c(u) = mu for all u ∈ V.
39. De Bruijn-Erd˝ os theorem Proof (Cont.) We claim that c is a chromatic coloring. To verify this, suppose u, v ∈ V and c(u) = c(v) = m.
40. De Bruijn-Erd˝ os theorem Proof (Cont.) We claim that c is a chromatic coloring. To verify this, suppose u, v ∈ V and c(u) = c(v) = m. Using the fact that U is an ultrafilter, find x ∈ Zu ∩ Zv such that cx (u) = m and cx (v) = m.
41. De Bruijn-Erd˝ os theorem Proof (Cont.) We claim that c is a chromatic coloring. To verify this, suppose u, v ∈ V and c(u) = c(v) = m. Using the fact that U is an ultrafilter, find x ∈ Zu ∩ Zv such that cx (u) = m and cx (v) = m. Since cx is a chromatic coloring of Gx , it follows that {u, v} / ∈ E.
42. De Bruijn-Erd˝ os theorem Proof (Cont.) We claim that c is a chromatic coloring. To verify this, suppose u, v ∈ V and c(u) = c(v) = m. Using the fact that U is an ultrafilter, find x ∈ Zu ∩ Zv such that cx (u) = m and cx (v) = m. Since cx is a chromatic coloring of Gx , it follows that {u, v} / ∈ E. Essentially the same proof yields the following theorem. Theorem Suppose κ is a strongly compact cardinal and χ < κ. If G is a graph such that every subgraph of G of size < κ has chromatic number ≤ χ, then Chr(G) ≤ χ.
43. Incompactness graphs Definition Suppose µ ≤ κ are cardinals and G is a graph.
44. Incompactness graphs Definition Suppose µ ≤ κ are cardinals and G is a graph. 1 G is (µ, κ)-chromatic if Chr(G) = κ and, for all subgraphs H that are smaller than G, we have Chr(H) ≤ µ.
45. Incompactness graphs Definition Suppose µ ≤ κ are cardinals and G is a graph. 1 G is (µ, κ)-chromatic if Chr(G) = κ and, for all subgraphs H that are smaller than G, we have Chr(H) ≤ µ. 2 G is (µ, κ)-coloring if Col(G) = κ and, for all subgraphs H that are smaller than G, we have Col(H) ≤ µ.
46. Incompactness graphs Definition Suppose µ ≤ κ are cardinals and G is a graph. 1 G is (µ, κ)-chromatic if Chr(G) = κ and, for all subgraphs H that are smaller than G, we have Chr(H) ≤ µ. 2 G is (µ, κ)-coloring if Col(G) = κ and, for all subgraphs H that are smaller than G, we have Col(H) ≤ µ. If µ < κ ≤ λ and µ+ < λ, then the existence of (µ, κ)-chromatic or (µ, κ)-coloring graphs of size λ is an instance of incompactness at λ.
47. Incompactness graphs Definition Suppose µ ≤ κ are cardinals and G is a graph. 1 G is (µ, κ)-chromatic if Chr(G) = κ and, for all subgraphs H that are smaller than G, we have Chr(H) ≤ µ. 2 G is (µ, κ)-coloring if Col(G) = κ and, for all subgraphs H that are smaller than G, we have Col(H) ≤ µ. If µ < κ ≤ λ and µ+ < λ, then the existence of (µ, κ)-chromatic or (µ, κ)-coloring graphs of size λ is an instance of incompactness at λ. Moreover, the larger the gap between µ and κ, the more extreme the instance of incompactness.
48. III. History
49. Historical results Chromatic incompactness 1 (Erd˝ os-Hajnal) If CH holds, there is an (ℵ0, ℵ1)-chromatic graph of size ℵ2.
50. Historical results Chromatic incompactness 1 (Erd˝ os-Hajnal) If CH holds, there is an (ℵ0, ℵ1)-chromatic graph of size ℵ2. 2 (Todorcevic) If κ is a regular, uncountable cardinal and there is a non-reflecting stationary subset of Sκ ℵ0 , then there is an (ℵ0, ≥ ℵ1)-chromatic graph of size κ.
51. Historical results Chromatic incompactness 1 (Erd˝ os-Hajnal) If CH holds, there is an (ℵ0, ℵ1)-chromatic graph of size ℵ2. 2 (Todorcevic) If κ is a regular, uncountable cardinal and there is a non-reflecting stationary subset of Sκ ℵ0 , then there is an (ℵ0, ≥ ℵ1)-chromatic graph of size κ. 3 (Todorcevic) If Martin’s Axiom holds, then there is an (ℵ0, 2ℵ0 )-chromatic graph of size 2ℵ0 .
52. Historical results Chromatic incompactness 1 (Erd˝ os-Hajnal) If CH holds, there is an (ℵ0, ℵ1)-chromatic graph of size ℵ2. 2 (Todorcevic) If κ is a regular, uncountable cardinal and there is a non-reflecting stationary subset of Sκ ℵ0 , then there is an (ℵ0, ≥ ℵ1)-chromatic graph of size κ. 3 (Todorcevic) If Martin’s Axiom holds, then there is an (ℵ0, 2ℵ0 )-chromatic graph of size 2ℵ0 . 4 (Shelah) If V = L and κ is a regular, non-weakly compact cardinal, then there is an (ℵ0, κ)-chromatic graph of size κ.
53. Historical results Chromatic incompactness 1 (Erd˝ os-Hajnal) If CH holds, there is an (ℵ0, ℵ1)-chromatic graph of size ℵ2. 2 (Todorcevic) If κ is a regular, uncountable cardinal and there is a non-reflecting stationary subset of Sκ ℵ0 , then there is an (ℵ0, ≥ ℵ1)-chromatic graph of size κ. 3 (Todorcevic) If Martin’s Axiom holds, then there is an (ℵ0, 2ℵ0 )-chromatic graph of size 2ℵ0 . 4 (Shelah) If V = L and κ is a regular, non-weakly compact cardinal, then there is an (ℵ0, κ)-chromatic graph of size κ. 5 (Rinot) If λ is an infinite cardinal and GCH and λ both hold, then there is an (ℵ0, ≥ λ)-chromatic graph of size λ+.
54. Historical results Chromatic compactness 1 de Bruijn-Erd˝ os Theorem
55. Historical results Chromatic compactness 1 de Bruijn-Erd˝ os Theorem 2 (Foreman-Laver) It is consistent with GCH that there is no (ℵ0, ℵ2)-chromatic graph of size ℵ2.
56. Historical results Chromatic compactness 1 de Bruijn-Erd˝ os Theorem 2 (Foreman-Laver) It is consistent with GCH that there is no (ℵ0, ℵ2)-chromatic graph of size ℵ2. 3 (Shelah) It is consistent with GCH that, for all 1 ≤ n < ω, there are no (ℵn, ≥ ℵn+1)-chromatic graphs of size ℵω+1.
57. Historical results Chromatic compactness 1 de Bruijn-Erd˝ os Theorem 2 (Foreman-Laver) It is consistent with GCH that there is no (ℵ0, ℵ2)-chromatic graph of size ℵ2. 3 (Shelah) It is consistent with GCH that, for all 1 ≤ n < ω, there are no (ℵn, ≥ ℵn+1)-chromatic graphs of size ℵω+1. Question Is it consistent that, for some “small” regular cardinal κ, there are no (ℵ0, ≥ ℵ1)-chromatic graphs of size κ?
58. Historical results Coloring incompactness 1 (Shelah) Suppose that µ < κ are infinite, regular cardinals and there is a non-reflecting stationary subset of Sκ µ . Then there is a (µ, µ+)-coloring graph of size κ.
59. Historical results Coloring incompactness 1 (Shelah) Suppose that µ < κ are infinite, regular cardinals and there is a non-reflecting stationary subset of Sκ µ . Then there is a (µ, µ+)-coloring graph of size κ. Sketch of proof. Let S ⊆ Sκ µ be a non-reflecting stationary set. For each α ∈ S, let Aα ⊆ α be a set of successor ordinals, cofinal in α, of order type µ.
60. Historical results Coloring incompactness 1 (Shelah) Suppose that µ < κ are infinite, regular cardinals and there is a non-reflecting stationary subset of Sκ µ . Then there is a (µ, µ+)-coloring graph of size κ. Sketch of proof. Let S ⊆ Sκ µ be a non-reflecting stationary set. For each α ∈ S, let Aα ⊆ α be a set of successor ordinals, cofinal in α, of order type µ. Define a graph G = (κ, E) by letting E = {{α, β} | α ∈ S, β ∈ Aα}.
61. Historical results Coloring incompactness 1 (Shelah) Suppose that µ < κ are infinite, regular cardinals and there is a non-reflecting stationary subset of Sκ µ . Then there is a (µ, µ+)-coloring graph of size κ. Sketch of proof. Let S ⊆ Sκ µ be a non-reflecting stationary set. For each α ∈ S, let Aα ⊆ α be a set of successor ordinals, cofinal in α, of order type µ. Define a graph G = (κ, E) by letting E = {{α, β} | α ∈ S, β ∈ Aα}. Question Are there consistently graphs exhibiting arbitrarily large incompactness gaps for the coloring number?
62. Historical results Coloring compactness 1 (Shelah) It is consistent that, for all infinite cardinals µ and κ > µ+, there are no (µ, ≥ µ+)-coloring graphs of size κ.
63. Historical results Coloring compactness 1 (Shelah) It is consistent that, for all infinite cardinals µ and κ > µ+, there are no (µ, ≥ µ+)-coloring graphs of size κ. 2 (Fuchino et al.) For any cardinal λ ≥ ℵ2, FRP(< λ) is equivalent to the assertion that whenever G is a graph of size < λ with uncountable coloring number, there is a subgraph of G of size and coloring number ℵ1.
64. Historical results Coloring compactness 1 (Shelah) It is consistent that, for all infinite cardinals µ and κ > µ+, there are no (µ, ≥ µ+)-coloring graphs of size κ. 2 (Fuchino et al.) For any cardinal λ ≥ ℵ2, FRP(< λ) is equivalent to the assertion that whenever G is a graph of size < λ with uncountable coloring number, there is a subgraph of G of size and coloring number ℵ1. 3 (Magidor-Shelah) Suppose µ < κ are infinite cardinals, with κ regular, and there is λ such that µ < λ ≤ κ and ∆λ,κ holds. Then there are no (µ, ≥ µ+)-coloring graphs of size κ.
65. Delta reflection Definition (Magidor-Shelah) Suppose λ ≤ κ are infinite cardinals, with κ regular. ∆λ,κ is the assertion that, for every stationary S ⊆ Sκ <λ and every algebra A on κ with fewer than λ relations, there is a subalgebra A of A such that otp(A ) < λ is a regular cardinal and S ∩ A is stationary in sup(A ).
66. Delta reflection Definition (Magidor-Shelah) Suppose λ ≤ κ are infinite cardinals, with κ regular. ∆λ,κ is the assertion that, for every stationary S ⊆ Sκ <λ and every algebra A on κ with fewer than λ relations, there is a subalgebra A of A such that otp(A ) < λ is a regular cardinal and S ∩ A is stationary in sup(A ). Theorem (Magidor-Shelah) It is consistent that ∆ℵ ω2 ,ℵ ω2+1 holds.
67. Delta reflection Definition (Magidor-Shelah) Suppose λ ≤ κ are infinite cardinals, with κ regular. ∆λ,κ is the assertion that, for every stationary S ⊆ Sκ <λ and every algebra A on κ with fewer than λ relations, there is a subalgebra A of A such that otp(A ) < λ is a regular cardinal and S ∩ A is stationary in sup(A ). Theorem (Magidor-Shelah) It is consistent that ∆ℵ ω2 ,ℵ ω2+1 holds. Question Does ∆λ,λ+ imply chromatic compactness at λ+? In particular, does it imply that there are no (ℵ0, ≥ ℵ1)-chromatic graphs of size λ+?
68. IV. New results, and some answers
69. Coloring incompactness Theorem Suppose µ and κ are infinite cardinals such that κ is regular and is not the successor of a cardinal of cofinality cf(µ), then there are no (µ, ≥ µ++)-coloring graphs of size κ.
70. Coloring incompactness Theorem Suppose µ and κ are infinite cardinals such that κ is regular and is not the successor of a cardinal of cofinality cf(µ), then there are no (µ, ≥ µ++)-coloring graphs of size κ. Corollary Suppose µ and λ are infinite cardinals and G is a (µ, λ)-coloring graph. Then λ ≤ µ++.
71. Coloring incompactness Theorem Suppose µ and κ are infinite cardinals such that κ is regular and is not the successor of a cardinal of cofinality cf(µ), then there are no (µ, ≥ µ++)-coloring graphs of size κ. Corollary Suppose µ and λ are infinite cardinals and G is a (µ, λ)-coloring graph. Then λ ≤ µ++. Question Is it consistent that there is an (ℵ0, ℵ2)-coloring graph of size ℵω+1?
72. (κ) Definition (Todorcevic) Suppose κ is a regular, uncountable cardinal. A (κ)-sequence is a sequence Cα | α < κ such that:
73. (κ) Definition (Todorcevic) Suppose κ is a regular, uncountable cardinal. A (κ)-sequence is a sequence Cα | α < κ such that: 1 for all α < κ, Cα is a club in α;
74. (κ) Definition (Todorcevic) Suppose κ is a regular, uncountable cardinal. A (κ)-sequence is a sequence Cα | α < κ such that: 1 for all α < κ, Cα is a club in α; 2 for all α < β < κ, if α ∈ acc(Cβ), then Cα = Cβ ∩ α;
75. (κ) Definition (Todorcevic) Suppose κ is a regular, uncountable cardinal. A (κ)-sequence is a sequence Cα | α < κ such that: 1 for all α < κ, Cα is a club in α; 2 for all α < β < κ, if α ∈ acc(Cβ), then Cα = Cβ ∩ α; 3 there is no club D in κ such that, for all α ∈ acc(D), we have Cα = D ∩ α.
76. (κ) Definition (Todorcevic) Suppose κ is a regular, uncountable cardinal. A (κ)-sequence is a sequence Cα | α < κ such that: 1 for all α < κ, Cα is a club in α; 2 for all α < β < κ, if α ∈ acc(Cβ), then Cα = Cβ ∩ α; 3 there is no club D in κ such that, for all α ∈ acc(D), we have Cα = D ∩ α. (κ) is the assertion that there is a (κ)-sequence.
77. (κ) Definition (Todorcevic) Suppose κ is a regular, uncountable cardinal. A (κ)-sequence is a sequence Cα | α < κ such that: 1 for all α < κ, Cα is a club in α; 2 for all α < β < κ, if α ∈ acc(Cβ), then Cα = Cβ ∩ α; 3 there is no club D in κ such that, for all α ∈ acc(D), we have Cα = D ∩ α. (κ) is the assertion that there is a (κ)-sequence. For infinite cardinals λ, (λ+) is a significant weakening of λ.
78. (κ) Definition (Todorcevic) Suppose κ is a regular, uncountable cardinal. A (κ)-sequence is a sequence Cα | α < κ such that: 1 for all α < κ, Cα is a club in α; 2 for all α < β < κ, if α ∈ acc(Cβ), then Cα = Cβ ∩ α; 3 there is no club D in κ such that, for all α ∈ acc(D), we have Cα = D ∩ α. (κ) is the assertion that there is a (κ)-sequence. For infinite cardinals λ, (λ+) is a significant weakening of λ. (κ) is manifestly an instance of incompactness, yet it is compatible with certain compactness principles (e.g., stationary reflection) holding at κ.
79. Chromatic incompactness Theorem If λ is an infinite cardinal and GCH and (λ+) both hold, then there is an (ℵ0, ≥ λ)-chromatic graph of size λ+.
80. Chromatic incompactness Theorem If λ is an infinite cardinal and GCH and (λ+) both hold, then there is an (ℵ0, ≥ λ)-chromatic graph of size λ+. Under certain additional assumptions (e.g., if λ is singular or some amount of stationary reflection holds at λ+), then the conclusion can be strengthened to the existence of an (ℵ0, λ+)-chromatic graph of size λ+.
81. Chromatic incompactness Theorem If λ is an infinite cardinal and GCH and (λ+) both hold, then there is an (ℵ0, ≥ λ)-chromatic graph of size λ+. Under certain additional assumptions (e.g., if λ is singular or some amount of stationary reflection holds at λ+), then the conclusion can be strengthened to the existence of an (ℵ0, λ+)-chromatic graph of size λ+. Theorem If κ is a regular, uncountable cardinal, then a generically-added (κ)-sequence gives rise to an (ℵ0, κ)-chromatic graph of size κ.
82. Corollaries Our techniques give rise to a number of corollaries indicating that large amounts of chromatic incompactness are compatible with significant compactness principles.
83. Corollaries Our techniques give rise to a number of corollaries indicating that large amounts of chromatic incompactness are compatible with significant compactness principles. In what follows, E(χ, κ) asserts the existence of a (χ, κ)-chromatic graph of size κ.
84. Corollaries Our techniques give rise to a number of corollaries indicating that large amounts of chromatic incompactness are compatible with significant compactness principles. In what follows, E(χ, κ) asserts the existence of a (χ, κ)-chromatic graph of size κ. Corollary The following are all consistent (relative to various large cardinal hypotheses).
85. Corollaries Our techniques give rise to a number of corollaries indicating that large amounts of chromatic incompactness are compatible with significant compactness principles. In what follows, E(χ, κ) asserts the existence of a (χ, κ)-chromatic graph of size κ. Corollary The following are all consistent (relative to various large cardinal hypotheses). 1 Stationary reflection at ℵω+1 together with E(ℵ0, ℵω+1).
86. Corollaries Our techniques give rise to a number of corollaries indicating that large amounts of chromatic incompactness are compatible with significant compactness principles. In what follows, E(χ, κ) asserts the existence of a (χ, κ)-chromatic graph of size κ. Corollary The following are all consistent (relative to various large cardinal hypotheses). 1 Stationary reflection at ℵω+1 together with E(ℵ0, ℵω+1). 2 ∆ℵ ω2 ,ℵ ω2+1 together with E(ℵ0, ℵω2+1 ).
87. Corollaries Our techniques give rise to a number of corollaries indicating that large amounts of chromatic incompactness are compatible with significant compactness principles. In what follows, E(χ, κ) asserts the existence of a (χ, κ)-chromatic graph of size κ. Corollary The following are all consistent (relative to various large cardinal hypotheses). 1 Stationary reflection at ℵω+1 together with E(ℵ0, ℵω+1). 2 ∆ℵ ω2 ,ℵ ω2+1 together with E(ℵ0, ℵω2+1 ). 3 χ is a supercompact cardinal together with E(χ, κ) for all regular κ > χ.
88. Corollaries Our techniques give rise to a number of corollaries indicating that large amounts of chromatic incompactness are compatible with significant compactness principles. In what follows, E(χ, κ) asserts the existence of a (χ, κ)-chromatic graph of size κ. Corollary The following are all consistent (relative to various large cardinal hypotheses). 1 Stationary reflection at ℵω+1 together with E(ℵ0, ℵω+1). 2 ∆ℵ ω2 ,ℵ ω2+1 together with E(ℵ0, ℵω2+1 ). 3 χ is a supercompact cardinal together with E(χ, κ) for all regular κ > χ. 4 Martin’s Maximum (or Rado’s Conjecture) together with E(ℵ2, κ) for all regular κ > ℵ2.
89. Speculative conclusions • These results provide further evidence that chromatic compactness at small cardinals, if consistent, is difficult to achieve, while compactness for the coloring number is much more tractable.
90. Speculative conclusions • These results provide further evidence that chromatic compactness at small cardinals, if consistent, is difficult to achieve, while compactness for the coloring number is much more tractable. • Recent work of Fuchino et al. has uncovered a number of compactness principles (in graph theory, Hilbert space theory, topology, etc.) that turn out to be equivalent to instances of FRP.
91. Speculative conclusions • These results provide further evidence that chromatic compactness at small cardinals, if consistent, is difficult to achieve, while compactness for the coloring number is much more tractable. • Recent work of Fuchino et al. has uncovered a number of compactness principles (in graph theory, Hilbert space theory, topology, etc.) that turn out to be equivalent to instances of FRP. • This raises the possibility that there may be interesting dividing lines between compactness principles that are “easy” to arrange at small cardinals (stationary reflection, compactness for countable coloring number, etc.) and those that seem “difficult” or even impossible (compactness for countable chromatic number, compactness for metrizability of first-countable topological spaces, etc.).
92. These results come from... Chris Lambie-Hanson and Assaf Rinot. Reflection on the coloring and chromatic numbers. Combinatorica, To appear, 2017.
93. Artwork by... Josef Albers (1888-1976)
94. Thank you!