peter komjath and saharon shelah- finite subgraphs of uncountably chromatic graphs

8/3/2019 Peter Komjath and Saharon Shelah- Finite subgraphs of uncountably chromatic graphs

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Finite subgraphs of uncountably chromatic

graphs

Peter Komjath

Saharon Shelah

October 6, 2003

Abstract

It is consistent that for every function f : there is a graphwith size and chromatic number 1 in which every n-chromatic subgraphcontains at least f(n) vertices (n 3). This solves a $ 250 problem ofErdos. It is consistent that there is a graph X with Chr(X) = |X| = 1such that if Y is a graph all whose finite subgraphs occur in X thenChr(Y) 2 (so the Taylor conjecture may fail). It is also consistent thatifX is a graph with chromatic number at least 2 then for every cardinal there exists a graph Y with Chr(Y) all whose finite subgraphs areinduced subgraphs ofX.

1 IntroductionIn [8] Erdos and Hajnal determined those finite graphs which appear as sub-graphs in every uncountably chromatic graph: the bipartite graphs. In fact, notjust that any odd circuit can be omitted, for every natural number n 1 andinfinite cardinal there is a graph with cardinality and chromatic number such that it omits all odd circuits up to length 2n + 1. They observed that theso-called r-shift graph construction has all but one of these properties; the vertexset of Shr() is the set of all r-tuples from , with {x0, x1, . . . , xr1}< joined to{x1, x2, . . . , xr}


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proved by Erdos, Hajnal, and Shelah [11], and independently, by Thomassen

[17].In [19] and later in [11] the following problem was posed (the Taylor conjecture). If, are uncountable cardinals and X is a -chromatic graph, isthere a -chromatic graph Y such that every finite subgraph of Y appears as asubgraph of X. Notice that the above shift graphs give some evidence for thisconjecturethe finite subgraphs of Shr() do not depend on the parameter .The authors of [11] remarked that even the following much stronger conjectureseemed possible. If X is uncountably chromatic, then for some r it contains allfinite subgraphs of Shr(). This conjecture was then disproved in [13].

One easy remark, the so called Hanf number argument gives that there is acardinal with the property that if the chromatic number of some graph X is atleast then there are arbitrarily large chromatic graphs with all finite subgraphsappearing in X. This argument, however, does not give any reasonable bound

on .Another conjecture of Erdos and Hajnal if the maximal chromatic number of

n-element subgraphs of an uncountably chromatic graph as a function ofn canconverge to infinity arbitarily slowly as n tends to infinity. It was mentionedin several problem papers, for example in [2], [5], [6], [10], [12]. See also [1],[14]. The relevance of the above examples is that the chromatic number of then-vertex subgraphs of (any) Shr() grows roughly as the r 1 times iteratedlogarithm ofn. Perhaps it was this fact that led Erd os and Hajnal to the aboveproblem.

Erdos also tirelessly popularized the Taylor conjecture, he mentioned it e.g.,in [3], [4], [7], [9], [10], [11]. It is also mentioned in [1], the book collectingErdos conjectures on graphs. In [15] we gave some results when the additional

hypotheses |X| = , |Y| = was imposed. We described countably manydifferent classes Kn,e of finite graphs and proved that if 0 = then every +-chromatic graph of cardinal + contains, for some n, e, all elements of Kn,e assubgraphs. On the other hand, it is consistent for every regular infinite cardinal that there is a +-chromatic graph on + that contains finite subgraphsonly from Kn,e. We got, therefore, some models of set theory, where the finitesubraphs of graphs with |X| = Chr(X) = + for regular uncountable cardinals were completely described.

Notice that the class of regular cardinals on which the above result operatedexcludes 1, and in this paper we show the reason, by resolving the aboveErdos-Hajnal conjecture: it is consistent that for every monotonically increasingfunction f : there is a graph with size and chromatic number 1 inwhich every n-chromatic subgraph has at least f(n) elements (n 3). Thepossibility of transforming the proof into a ZFC argument will be checked inthe forthcoming [16], Chapter 9. An application of the method presented heregives the consistent existence of a graph X with Chr(X) = |X| = 1 suchthat if Y is a graph (of any size) all whose subgraphs are subgraphs of X thenChr(Y) 2. This gives a consistent negative answer to the Taylor conjecture.As for the positive direction we prove that it is consistent that if X is a graphwith chromatic number at least 2 then there are arbitrarily large chromatic

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graphs all whose finite subgraphs being induced subgraphs of X.

Theorems 1 and 2 were proved by S. Shelah and then P. Komjath provedTheorems 3 and 4.

Notation. We use the standard axiomatic set theory notation. If A is a setof ordinals, is an ordinal, then < A, means that < holds for every A. Similarly for A, A < , etc. If f is a function, A a set, then we letf[A] = {f(x) : x A}. If S is a set, is a cardinal, [S] = {X S : |X| = },[S]


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We notice that although a condition p = (s,u,g,m,hi, c) is infinite (perhaps

p = (s,u,g,m, (hi)i


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For (3.) assume that C is an odd circuit of length f(r) in Yp

r . If we

replace every e C that contains at least one vertex from s s

with (e) thenwe get an odd cycle C in Yp

r . C splits into circuits, at least one of them odd,

so we get a contradiction.For (2.) notice that it holds if x, y are joined via a path going entirely in Ypr

or Yp

r . It suffices, therefore, to show that if some path P between x and y oflength i is split by an inner point z into the paths P0 and P1 between x and z,and z and y, respectively, and of the respective lengths i0 and i1 (so i0 + i1 = i)and the statement holds for P0 and P1 then it holds for P, as well. Indeed, fori j f(r) i we have

hj(i0+i1)(x) < hji1(z) < hj(y) < hj+i1(z) < hj+(i0+i1)(x).

Lemma 5. Chr(X) = 1.

Proof. Assume that some p forces that F is a good coloring of X with theelements of. Select an increasing, continuous sequence of countable elementarysubmodels p , F , Qf, N0 N1 N H() with some large enoughregular cardinal , for < 1, such that = N 1 is an ordinal. The setC = { : < 1} will be a closed, unbounded set, so by the guessing propertythere is some = such that C C. Notice that all points of p are smallerthan .

Extend p to a p using Lemma 2., adding to the u-part, then let p be acondition extending p such that p F() = i holds for some i < .

Let n be some natural number that n / {c(x, ) : {x, } g, x < } wherep = (s, u, g, m, h

i

, c). Extend p using Lemma 3., to some condition pwith p = (s,u,g,m,hi, c) such that m = m() f(n) holds.

In p we have the values

h0() < h1() < < hm1() <

and by our requirements on conditions there are elements 0, . . . , m1 of Csuch that

0 < h0() < 1 < h1() < < m1 < hm1() <

holds.The values 0, . . . , m1, break s into disjoint parts: s = s0 smsm+1

with

s0 < 0 s1 < 1 s2 < < m1 sm < sm+1

(some of them may be empty).

Sublemma. There is a condition p on some s = s0 s1 sm s

m+1

isomorphic to p with |si| = |si|, = min(sm+1), p

F() = i and

s0 < s

1 < 0 s1 < s

2 < 1 s2 < < s

m < m1 sm < s

m+1 < sm+1

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holds.

Proof. Let be the isomorphism type of p. For the ordered finite setsx0, x1, . . . , xm+1 let (x0, x1, . . . , xm+1) denote the statement that x0 < x1 < < xm+1, |xi| = |si| and for the (unique) condition p on x0 x1 xm+1of type p F() = i where = min(xm+1).

Setm+1(x0, x1, . . . , xm+1) = (x0, x1, . . . , xm+1)

and for 0 i m

i(x0, x1, . . . , xi) = xi+1i+1(x0, x1, . . . , xi+1)

where the quantifier denotes there exist unboundedly many which is ex-pressible in the first order language of (1,


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As p forces that F() = F() = i yet they are joined in X, we are finished.

Theorem 1. The forcing Qf adds an uncountably chromatic graph X on 1such that every subgraph on at most f(r) vertices is at most 2r+1-chromatic.

Proof. As every Xn is a circuitfree graph, it can be colored with two colors.Consider now a subgraph of X induced by a set S of at most f(r) vertices. OnS all the graphs X0, . . . , X r1 are bipartite, and so is Yr = Xr (as it hasno odd circuits of length f(r)). So their union, X restricted to S, can becolored by at most 2r+1 colors.

Theorem 2. It is consistent with CH that for every function f : thereis an uncountably chromatic graph X on 1 such that every sugraph of X onf(r) vertices is at most r-chromatic (r 2).

Proof. Assume that holds in the ground model. Then we have CH and thereis a club guessing sequence {C : < 1, limit} as required for Theorem 1. Weforce with a finite support iteration P = {P, Q : < 1}. In step < 1we add Q = Qf for some increasing function f : . As this will be accc forcing that preserves CH it is possible by bookkeeping to make sure thatevery suitable f : occurs as some f. Also, as P, the iteration up to is ccc, every closed, unbounded set C in VP contains a ground model closed,unbounded set D, and as there is some element C of the club guessing systemthat C D we have C C, that is the club guessing system retains itsproperty in VP .

Call a condition p P determined if for every < 1 the condition p|completely determines p(), that is, for every coordinate p() is not just a

name for a finite structure but it is actually a finite structure.Lemma 6. The determined conditions form a dense set in P.

Proof. We prove by induction on < 1 that the determined conditions forma dense subset of P. This is obvious if is limit, as we are considering finitesupports. Assume that we have the statement for some < 1 and we try tohandle the case of + 1. Let (p,q) P+1 = P Q be arbitrary. Extendp to some p that completely determines q, that is, there is a finite structure hthat p q = h. Then extend p to a determined p P. Now (p, h) is adetermined extension of (p,q).

Lemma 7. For every < 1, Chr(X) = 1 holds in VP.

Proof. By moving to VP we can assume that = 0. We imitate the proof ofLemma 5. By Lemma 6 we can work with determined conditions. We considerevery such condition as a finite structure on some finite subset s of 1, heres contains all points of all graphs p() where is an arbitrary element of thesupport of p, and we also add the elements of the support to s. Assume thatsome p P forces that F is a good coloring ofX0 with the elements of . Withan argument like in Lemma 5 we get some natural number n, ordinals < . By the choice of , there is a graph Z with Chr(Z) such that everyfinite induced subgraph of Z is an induced subgraph of Y.

Lemma 10. Chr(Z) holds in V[G].

Proof. Otherwise let F be a name for a coloring with the ordinals less than < . Then the coloring x (p,) is a good coloring of the vertices of Z with + < colors, where p P is some element of P with p F(x) = .

We are almost finished, the only problem is that the finite induced subgraphsofZ are not induced subgraphs ofX, they only are (edge-)subgraphs of inducedsubgraphs of X. The following Lemma is what we need.

Lemma 11. There is a graph Z on the vertex set of Z with Z Z and suchthat every induced subgraph of Z is an induced subgraph of X.

Proof. Let S be the vertex set of Z. For every finite subset s of S there aresome graphs on s, which on the one hand are isomorphic to induced subgraphsofX, on the other hand they are supergraphs of the graph Z restricted to s. Callthese graphs appropriate for s. Notice that there are finitely many appropriategraphs for every given s, and ifT is an appropriate graph for s and s is a subsetofs then T restricted to s is a graph appropriate for s. We can therefore applythe Rado selection principle (or the compactness theorem of model theory) andget a graph Z on S every induced subgraph of which is appropriate, so Z is arequired.

As Chr(Z) Chr(Z) holds we are done.

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References

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[3] P. Erdos: On the combinatorial problems which I would most like to seesolved, Combinatorica 1 (1981), 2542.

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[5] P. Erdos: Some of my favourite unsolved problems, A tribute to Paul Erdos,Cambridge Univ. Press, Cambridge, 1990., 467478.

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[9] P. Erdos, A. Hajnal: Unsolved and solved problems in set theory, Proceed-ings of the Tarski Symposium (Proc. Symp. Pure Math., Vol. XXV, Univ.of California, Berkeley, Calif., 1971), Amer.Math.Soc., Providence, R.I.,1974, 269287.

[10] P. Erdos, A. Hajnal: Chromatic number of finite and infinite graphs andhypergraphs. (French summary) Special volume on ordered sets and theirapplications (LArbresle, 1982). Discrete Math. 53(1985), 281285.

[11] P.Erdos, A.Hajnal, S.Shelah: On some general properties of chromaticnumber, in: Topics in Topology, Keszthely (Hungary), 1972, Coll. Math.

Soc. J. Bolyai 8, 243255.

[12] P. Erdos, A. Hajnal, E. Szemeredi: On almost bipartite large chromaticgraphs, Annals of Discrete Math. 12(1982), 117123.

[13] A. Hajnal, P.Komjath: What must and what need not be contained in agraph of uncountable chromatic number ? Combinatorica4 (1984), 4752.

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[14] T. R. Jensen, B. Toft: Graph coloring problems, John Wiley and Sons,

1995.[15] P.Komjath, S. Shelah: On Taylors problem, Acta Math. Hung., 70 (1996),

217225.

[16] S. Shelah: Non-structure Theory, Oxford University Press, to appear.

[17] C. Thomassen: Cycles in graphs of uncountable chromatic number, Com-binatorica 3 (1983), 133134.

[18] W.Taylor: Atomic compactness and elementary equivalence, Fund. Math.71 (1971), 103112.

[19] W.Taylor: Problem 42, Comb. Structures and their applications, Proc. ofthe Calgary International Conference, 1969.

Peter KomjathDepartment of Computer ScienceEotvos UniversityBudapest, P.O.Box 1201518, Hungarye-mail: [email protected]

Saharon ShelahInstitute of Mathematics,Hebrew University,Givat Ram, 91904,Jerusalem, Israele-mail: [email protected]

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peter komjath and saharon shelah- finite subgraphs of uncountably chromatic graphs

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