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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
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‘The purest and most thoughtful minds are
those which love colour the most.’
-John Ruskin
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I. Chromatic and coloring numbers
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Graphs
Definition
1 A graph is a pair G = (V, E) such that E ⊆ [V]2.
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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}.
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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}.
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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).
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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.
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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.
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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}.
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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.
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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.
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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).
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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).
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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).
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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).
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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).
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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).
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II. Reflection/Compactness
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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].
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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].
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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.
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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.
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Compactness heuristics
• ℵ0 displays a high degree of compactness:
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Compactness heuristics
• ℵ0 displays a high degree of compactness:
Compactness theorem for first-order logic
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Compactness heuristics
• ℵ0 displays a high degree of compactness:
Compactness theorem for first-order logic
K¨
onig’s infinity lemma
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Compactness heuristics
• ℵ0 displays a high degree of compactness:
Compactness theorem for first-order logic
K¨
onig’s infinity lemma
Ramsey’s theorem
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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:
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Incompactness graphs
Definition
Suppose µ ≤ κ are cardinals and G is a graph.
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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.
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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.
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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.
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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.
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III. History
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Historical results
Chromatic incompactness
1 (Erd˝
os-Hajnal) If CH holds, there is an (ℵ0, ℵ1)-chromatic
graph of size ℵ2.
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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.
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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 .
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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.
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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+.
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Historical results
Chromatic compactness
1 de Bruijn-Erd˝
os Theorem
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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.
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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.
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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?
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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.
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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.
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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}.
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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?
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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.
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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.
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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.
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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 ).
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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.
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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+?
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IV. New results, and some answers
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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.
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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++.
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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?
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(κ)
Definition (Todorcevic)
Suppose κ is a regular, uncountable cardinal. A (κ)-sequence is
a sequence Cα | α < κ such that:
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(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;
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(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;
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(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.
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(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.
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(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.
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(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.
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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+.
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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+.
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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.
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Corollaries
Our techniques give rise to a number of corollaries indicating that
large amounts of chromatic incompactness are compatible with
significant compactness principles.
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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.
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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).
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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).
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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
).
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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.
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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.
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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.
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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.
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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.).
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These results come from...
Chris Lambie-Hanson and Assaf Rinot. Reflection on the coloring
and chromatic numbers. Combinatorica, To appear, 2017.
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Artwork by...
Josef Albers (1888-1976)
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Thank you!