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.
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.
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.
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.
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).
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).
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).
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).
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).
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).
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].
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.
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.
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
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 .
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.
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+.
(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.
(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?
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.
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}.
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?
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.
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.
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 ).
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.
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+?
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++.
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?
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.
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.
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.
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.
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+.
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.
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.
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).
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).
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 ).
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.
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.
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.
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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