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Finite subgraphs of uncountable graphs
Chris Lambie-Hanson
Department of Mathematics and Applied Mathematics
Virginia Commonwealth University
CMU Mathematical Logic Seminar
28 April 2020
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I. Graph theoretic background
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Basic definitions
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Basic definitions
Definition
• A graph is a pair G = (Σ, E), where E ⊆ [Σ]2.
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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.
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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.
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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.
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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:
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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;
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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;
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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.
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De Bruijn-Erd˝
os Theorem
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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.
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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.
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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.
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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.
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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?
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Previous work
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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 .
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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
.
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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 ?
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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 .
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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.
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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.
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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.
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Current status
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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 .
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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.
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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.
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II. Type graphs
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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.
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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}.
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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}.
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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.
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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.
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Type graphs
Definition
Suppose that n is natural number, t is a disjoint type of width n,
and α is an ordinal.
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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.
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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.
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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|.
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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.
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A special case
G(α, t3
1
) is triangle-free.
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A special case
G(α, t3
1
) is triangle-free.
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A special case
G(α, t3
1
) is triangle-free.
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A special case
G(α, t3
1
) is triangle-free.
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A special case
G(α, t3
1
) is triangle-free.
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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 ).
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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.
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III. The main construction
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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.
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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.
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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.
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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.
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The vertices
Σ is the set of all triples σ = Mσ, fσ, Cσ such that
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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;
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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;
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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.
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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 .
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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.
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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.
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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.
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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.)
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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
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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}, . . ..
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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 }.
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The edges
When constructing E, we will observe the following constraints.
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The edges
When constructing E, we will observe the following constraints.
• If {σ, τ} ∈ E, then σ ≺ τ or τ ≺ σ.
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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.
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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.
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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
.
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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.
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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.
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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).
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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).
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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).
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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 .
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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.
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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
).
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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.
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Finite subgraphs
Fix k ∈ N and a subgraph H of G with at most f (k) vertices.
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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
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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
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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
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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
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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
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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
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Chromatic number
It remains to show that χ(G) = ℵ1.
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Chromatic number
It remains to show that χ(G) = ℵ1. χ(G) ≤ ℵ1 is easy.
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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.
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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 ).
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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.
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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 .
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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).
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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.
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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.
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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 . . .”.
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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.
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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}),
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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+).
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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.
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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}).
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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}).
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Moving back up
We have
∃∞γ0 ∃∞γ1 ∃∞γ2 ∃σ (f (σ) = k ∧ Cσ[3] = {γ0, γ1, γ2}).
and t = t3
1
= 001011.
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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).
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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
}.
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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
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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.
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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
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IV. Further questions
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On optimality
Two natural questions arise regarding the optimality of our main
theorem.
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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 ?
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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 ?
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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?
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Other questions
We end with some other questions of Erd˝
os and Hajnal about
finite subgraphs of uncountably chromatic graphs.
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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?
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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?
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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?
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References
Erd˝
os, Paul. Some of my favourite unsolved problems. A tribute
to Paul Erd˝
os, 467–478, Cambridge Univ. Press, Cambridge,
1990.
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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
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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.
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Thank you!