Péter Komjáth

Coloring graphs with locally few colors

Abstract Let G be a graph, m > r ⩾1 integers. Suppose that it has a good coloring with m colors which uses at most r colors in the neighborhood of every vertex. We investigate these so-called local r -colorings. One of our results (Theorem 2.4) states: The chromatic number of G , Chr( G )⩽ r 2 r log 2 log 2 m (and this value is the best possible in a certain sense). We consider infinite graphs as well.

Some universal graphs

It is shown that various classes of graphs have universal elements. In particular, for eachn the class of graphs omitting all paths of lengthn and the class of graphs omitting all circuits of length at leastn possess universal elements in all infinite powers.

There is no universal countable pentagon-free graph

No countable Cn-free graph contains every countable Cn-free graph as a subgraph, for n ≧ 4. For n = 4, this was proved earlier by J. Pach.

Universal elements and the complexity of certain classes of infinite graphs

A class of graphs has a universal element G0, if every other element of the class is isomorphic to an induced subgraph of G0. In Sections 1?4 we give a survey of some recent developments in the theory of universal graphs in the following areas: (1) Graphs universal for isometric embeddings, (2) universal random graphs, (3) universal graphs with forbidden subgraphs, (4) universal graphs with forbidden topological subgraphs. Section 5 is devoted to the problem of deciding how far a class of graphs G is from having a universal element. We introduce a new measure of the complexity of the class G, denoted by cp(G). This is defined to be the minimum cardinal ? such that there exist ? elements in G with the property that any other element of G can be embedded into at least one of them as an induced subgraph. G has a universal element if and only if cp(G)=1. Among other theorems we prove that (i) the complexity of the class of all countable graphs without n ? 2 independent edges is finite; (ii) for any cardinal G, ?1???2?, it is consistent that the complexity of the class of all locally finite countable graphs is equal to ?. In Section 6 we consider some analogous questions for hypergraphs.

Coloring of universal graphs

LetG be a graph andr a cardinal number. Extending the theorem of J. Folkman we show that if eitherr or clG are finite then there existsH with clH = clG andH → (G)r1. Answering a question of A. Hajnal we show that countably universal graphU3 satisfiesU3 → (U3)r1 for every finiter.

THE CLUB GUESSING IDEAL: COMMENTARY ON A THEOREM OF GITIK AND SHELAH

It is shown in this paper that it is consistent (relative to almost huge cardinals) for various club guessing ideals to be saturated.

Some remarks on universal graphs

Abstract A corrected proof is given for the existence of a universal countable {C3, C5, …, C2s+1}-free graph. We also prove that there is a universal countable free graph. There is no universal countable H-free graph if H is the dispoint union of 3 or more complete n-cliques for some n ⩾ 2, plus one vertex, joined to every other point.

Embedding graphs into colored graphs

If X is a graph, K a cardinal, then there is a graph Y such that if the vertex set of Y is s-colored, then there exists a monocolored induced copy of X; moreover, if X does not contain a complete graph on ce vertices, neither does Y. This may not be true, if we exclude noncomplete graphs as subgraphs. It is consistent that there exists a graph X such that for every graph Y there is a two-coloring of the edges of Y such that there is no monoeolored induced copy of X. Similarly, a triangle-free X may exist sueh that every Y must contain an infinite complete graph, assuming that coloring Ys edges with countably many colors a monocolored copy of X always exists. 0. Introduction. In this paper we deal with the generalization of partition theory which investigates the existence of monocolored prescribed subgraphs of multicolored graphs satisfying certain conditions. As usual we will need partition symbols to make the formulation of the results and problems feasible. (0.1) Y ) (X)l X y > (X)2 mean that the following statements are true. If the vertices/edges of Y are ey-colored then there exists a monocolored copy of X c Y, respectively. (0.2) Y (X)1, Y (X)2 mean the existence of monocolored copies of X which are induced subgraphs of Y. Clearly, the Erdos-Rado generalization of Ramseys theorem yields an obvious existence theorem of type VX BY in (0.1), and the meaningful results concerning this symbol are of the form VX E BY E g for certain classes , g of graphs. The existence problem for the symbols (0.2) is nontrivial, though it is quite easy for the first symbol and here the problem has to be investigated under additional restrictions on X and Y. As to t symbol Y (X)2 the stat m nt VX BY Y (X)2 forey < ,X and Y) finite was proved by three diderent sets of authors [4, 10, 20] and it was extended for countable graphs X in [10] wher Vt W V0Xt < W HtY < 2w y (X)2 was proved. One of the main observations of this paper is that (contrary to the intuitive expectation of combinatorialists that this kind of Ramsey property always holds Received by the editors March 21, 1986 and, in revised form, April 15, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 03E05; Secondary 03E35, 04A20, 05C65.

A simplified construction of nonlinear Davenport—Schinzel sequences

Abstract A simplified construction for a nonlinear Davenport-Schinzel sequence is given. This proves λ 2s + 1 (n) = Ω(nα s (n)) .

What must and what need not be contained in a graph of uncountable chromatic number

We investigate the following problem: What countable graphs must a graph of uncountable chromatic number contain? We define two graphsΓ andΔ which are very similar and we show thatΓ is contained in every graph of uncountable chromatic number, whileΔ is (consistently) not.