bounding colorability upper bounds - bgukojman/owtalk.pdf · introduction bounding colorability...

IntroductionBounding colorability

Upper bounds

Bounds on coloring numbers

Menachem Kojman

Ben-Gurion University, Beer Sheva, and the Institute for Advanced Study,Princeton NJ

January 15, 2011

Menachem Kojman Bounds on coloring numbers


Upper bounds

Table of contents

1 Introduction

2 Bounding colorability

3 Upper boundsInfinite list-chromatic numberAssuming cardinal arithmetic is tameIn ZFC with the revised power function



Upper bounds

Basic definitions

The chromatic number of a graph G = (V ,E ) is the leastcardinal κ for which there exists a proper vertex coloringc : V → κ of G .

The list-chromatic or choice number χ`(G ) is a variation onthe chromatic number in which each vertex v ∈ V is assignedits own list of colors L(v) and the proper coloring choosesc(v) ∈ L(v).

χ`(G ) is the least cardinal κ such that for every assignment oflists L(v) of size |L(v)| = κ for each v ∈ V there exists achoice function c which is a proper coloring of the graph.


Figure: χ(K (3, 3)) = 2, χ`(K (3, 3)) = 3


Upper bounds

For a bipartite graph K (κ, λ) the list-chromatic number is atmost (min{κ, λ})+.

If m ≥ nn then χ`(K (n,m) = n + 1


Figure: χ(K (3, 3)) = 2, χ`(K (3, 3)) = 3

Figure: χ(K (n,m)) = n + 1

Figure: χ(K (3, 3)) = 2, χ`(K (3, 3)) = 3


Upper bounds

The coloring number Col(G ) is the least cardinal κ for whichthere exists a well ordering ≺ of V such that|G≺[v ]| = |{u : u ≺ v ∧ {u, v} ∈ E}| < κ for every v ∈ V .

For every graph G , the coloring number is at most1 + max{deg(v) : v ∈ V } ≤ |V |. The coloring number ofK (3, 3) is 4.


Figure: Col(K (3, 3)) = 4


Upper bounds

Summary of definitions

For every graph G ,

χ(G ) ≤ χ`(G ) ≤ Col(G ).

and the inequalities may be strict.



Upper bounds

Some History

The list-chromatic number was introduced independently byVizing in 1976 and Erdos, Rubin and Taylor in 1979 and thenlay dormant for a long time. Since the 1990s this numberattracts a lot of interest in the graph theory community.

The coloring number was introduced by Erdos and Hajnal intheir work on graphs of uncountable chromatic number in1966 (or earlier?). They observed that some of their resultsremained valid in the broader class of graphs with uncountablecoloring number.

Recently some interest has been given to list-chromaticnumbers of relatives of the unit distance graph on R2.



Upper bounds

Alon’s result and question

Let d = d(G ) denote the minimum degree of a vertex in G .

In a finite graph, Col(G ) ≥ d(G ), because, as mentionedearlier, some vertex has to be the last in every ordering of thegraph. The λ-branching tree of height ω has uniform degree λbut has colorability 2.

In 2000 Alon proved that d(G ) ≤ (4 + ε)χ`(G) for every finitegraph G , using the probabilistic method.Given a finite G , finda vertex v with deg(V ) ≤ (4 + ε)χ`(G) and mark it as the lastvertex. Then eliminate this vertex from the graph andcontinue inductively. Thus:

Col(G ) ≤ (4 + ε)χ`(G).

Question (Alon): is there a similar bound on Col(G ) forinfinite graphs?



Upper bounds

1 The bipartite graph Kκ,κ has chromatic number 2, coloringnumber 1 + κ and in the finite case χ`(K (n, n)) grows toinfinity with n.

2 χ`(K (ℵ0, 2ℵ0)) = ℵ1 and more generally, χ`(K (κ, 2κ)) = κ+.

3 If κ < 2ℵ0 then χ`(K (ℵ0, κ)) = ℵ0.



Upper bounds

Komjath’s consistency results

MA implies that for every graph |G | < 2ℵ0 with countablechromatic number, χ`(G ) = ℵ0. By this result, there is noupper bound on coloring numbers in terms of list-chromaticnumbers using only the ℵ function; some exponentiation isneeded for a ZFC bound.

It is consistent that ℵ1 < 2ℵ0 and that there exists a graphG = (ω1,E ) with countable chromatic number andχ`(G ) = ℵ1.

It is consistent with the GCH that

χ`(G ) = ℵ0 =⇒ Col(G ) = ℵ0

for every graph G .

It is consistent with the GCH to have a graph withχ`(G ) = ℵ0 < Col(G ) = ℵ1.



Upper bounds

Infinite list-chromatic numberAssuming cardinal arithmetic is tameIn ZFC with the revised power function

Bounding coloring numbers inductively

Erdos and Hajnal introduced in 1966 a natural scheme forbounding colorability inductively: partition V = {Ci : i < θ} with|Ci | < |V | and use the induction hypothesis to well-order each Ci

separately. If the partition can be found so that G [v ] ∩⋃

j<i Cj isbounded for v ∈ Ci , then we are done.



Upper bounds


Their idea was to use the assumption that G is K (n, ω1)-free tofind sets Ci which are “closed” under common neighbors ofn-element sets.



Upper bounds


Closure operation

For every G = (V ,E ) let F : P(V )→ P(V ) be the functionF (X ) =

⋂v∈X G [v ] associating to X the set of common

neighbors of all vertices in X . For a cardinal κ letFκ = F � [V ]κ.

F and each Fκ are anti-monotone: X ⊆ Y =⇒ F (Y ) ⊆ F (X ).

A ⊆ V is κ-closed if F (X ) ⊆ A for every X ∈ [A]κ.

If A is κ-closed then |G [v ] ∩ A| < κ for all v ∈ V \ A.



Upper bounds


The finitary case: Erdos-Hajnal

Theorem (Erdos-Hajnal 1966)

Suppose G is K (n, ω1)-free for some n. Then Col(G ) ≤ ℵ0.

Corollary

For every graph G , if χ`(G ) < ℵ0 then Col(G ) ≤ ℵ0.

Proof.

Suppose χ`(G ) = n. Then G is K (M,M)-free for some finite M byAlon’s theorem or by our direct counting. Now apply the inductivescheme for FM -closed sets.



Upper bounds


Getting started

Assume now that χ`(G ) = κ is infinite. G is K (κ, 2κ)-free, and weshall try to work with κ-closure. So here are additional definitionsand properties of κ-closed sets for an infinite κ:

1 A cardinal θ is κ-stable for a graph G if every set A ∈ [V ]θ iscontained in a κ-closed set of the same cardinality.

2 If θκ = θ then θ is κ-stable for any G which is K (κ, 2κ)-free.

3 If {Bi : i < θ} is ⊆-increasing, each Bi is κ-closed andcf θ 6= cf κ, then

⋃i<θ Bi is κ-closed.

Proof of (3).

Just the case θ < cf κ. For every X ∈ [⋃

Bi ]κ there is some i < θ

such that |X ∩ Bi | = κ. Now use anti-monotonicity.



Upper bounds


Set κ = ℵ0 for the moment. Assume |V | = λ = (2ℵ0)+ and G isK (ℵ0, 2

ℵ0)-free. Let V =⋃

i<λ Bi , an increasing union with|Bi | = 2ℵ0 and ℵ0-closed. To make a set closed iterate ω1 timesthe operation A 7→ A ∪

⋃{F (X ) : X ∈ [A]ℵ0}. Now let

I = {i < λ : Bi \⋃

j<i Bj 6= ∅} and put Ci = Bi \⋃

j<i Bj for i ∈ I .This is a partition of V ; if cf i 6= ω then

⋃j<i Bj is ℵ0-closed, so a

vertex v ∈ Cj will have a finite set of neighbors in this union. Ifcf i = ω then v ∈ Ci may have ≤ ℵ0 neighbors in this union. Sowe are proving inductively that Col(G ) ≤ ℵ1.Similar for λ = (2ℵ0)+n.Similar for the first limit above 2ℵ0 .What about the successor of the first limit above 2ℵ0?



Upper bounds


Limit of countable cofinality

Pay more. Use the weaker ℵ1-closure operation. Recall that:

Recall:

A countable union of ℵ1-closed sets is ℵ1-closed.

Thus we can get ℵ1-closed sets, but then for |V | = ℵω+1 we onlyget Col(G ) ≤ ℵ2, because of limits of cofinality ℵ1 in a filtration toclosed sets.So we can pass every limit cardinal of cofinality ℵ0.This gets us as far as the first limit of cofinality ω1; what next?Settle for ℵ2? And then what?



Upper bounds


With weak SCH

Assume that every limit µ > 2ℵ0 of cofinality ω1 is closed underℵ0-exponentiation, that is, θ < µ⇒ θℵ0 < µ.

Lemma

Every cardinal θ ≥ 2ℵ0 is ℵ1-stable for all K (ℵ0, 2ℵ0)-free G .

Proof.

By induction on θ ≥ 2ℵ0 . Every limit of cofinality ω0 maintains theinduction hypothesis and limits of cofinality ω1 are limits ofℵ0-stable cardinals, so are even ℵ0-stable.



Upper bounds


Theorem

Col(G ) ≤ max{2ℵ0 ,ℵ2} for all G with χ`(G ) = ℵ0.

Proof.

Assume for simplicity 2ℵ0 = ℵ2.By induction on |V | = λ ≥ 2ℵ0 prove that Col(G ) ≤ ℵ2 for everyK (ℵ0, 2

ℵ0)-free G .Case 1. cf λ = ℵ1. Fix 〈θi : i < ω1〉 increasing with limit λ suchthat θℵ0

i = θi . Possible by the assumption. Present V =⋃

i<ω1Bi ,

increasing union, where |Bi | = θi and Bi is ℵ0-closed. LetI = {i < ω1 : Bi \

⋃j<i Bj 6= ∅} and Ci = Bi \

⋃j<i Bj for i ∈ I .

Case 2. cf λ 6= ℵ1 — even easier.



Upper bounds


Silver, Prikry and Gitik

Recall that modulo large cardinals it is consistent to have µℵ0

arbitrarily large for a strong limit µ of cofinality ω. Gitik provedthis for the first fixed point. In particular, for every µ′ ∈ (µ, µℵ0) itholds that µcf µ ≥ µℵ0 , so arbitrary high cofinalities may show up.

Thus, a simple counting argument using standard exponentiationwill not work in ZFC.

But we are not restricted to using only cardinal-arithmetic functionswhich were created in the limited world of natural number.



Upper bounds


With no assumptions

Shelah’s revised power function:

λ[κ] = min{|A| : A ⊆ [λ]κ ∧ (∀X ∈ [λ]κ)(∃Y ∈ [A]<κ)(X ⊆⋃Y)}

Lemma

If θ ≥ 2κ, κ = cf κ and θ[κ] = θ then θ is κ-stable for everyK (κ, θ+)-free G .

Proof.

Suppose A ∈ [V ]θ. Fix A ⊆ [A]κ witnessing θ[κ] = θ.⋃X∈[A]κ

F (X ) =⋃

Z∈A

⋃W∈[Z ]κ

F (W )

Iterate κ+ times the operation A 7→ A∪⋃{{F (X ) : X ∈ [A]κ}.



Upper bounds


Shelah’s revised GCH in ZFC:

For every λ ≥ iω(ν) for all but a bounded set of κ < iω(ν)

λ[κ] = λ.

Lemma

For every cardinal ν, every θ ≥ iω(ν) is κ-stable for all but abounded set of κ < iω(ν) for every K (ν, 2ν)-free G .

Proof.

Fix a regular κ < iω(ν) for which θ[κ] (there is one by Shelah’stheorem). Clearly 2κ < θ. κ-stability of θ follows by the previouslemma and persists upwards with κ.



Upper bounds


In ZFC

Theorem

Col(G ) ≤ iω(ν) for every graph G with χ`(G ) = ν.

Proof.

For every θ ≥ iω(ν) fix κ(θ) ∈ {(in(ν))+ : n ∈ ω} for such that θis κ(θ)-stable for every K (ℵ0, 2

ℵ0)-free G .Case 1: cf λ = ω. Let V =

⋃Bn, increasing union Bn is κ(|Bn|)

closed.Case 2: cf λ > ω. Fix a ≤-increasing sequence 〈θi : i < cf λ〉unbounded below λ and assume, without loss of generality, thatκ(θi ) is fixed. Let V =

⋃i<cf λ Bi , each Bi κ-closed. Now at limits

of cofinality other than κ the union is closed, and at limits ofcofinality exactly κ the trace of every vertex from outside is≤ κ.



Upper bounds


Last remark

The case ν ∈ ω is also included in the previous theorem by thecorollary to the theorem by Erdos and Hajnal we stated above, asiω(n) = ω.And now

Figure: coffee break


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