On Almost Bipartite Large Chromatic Graphs

In the past we have published quite a few papers on chromatic numbers of graphs (finite or infinite), we give a list of those which are relevant to our present subject in the references . In this paper we will mainly deal with problems of the following type : Assuming that the chromatic number x(1) of a graph is greater than K, a finite or infinite cardinal, what can be said about the behaviour of the set of all finite subgraphs of 9 . We will investigate this problem in case some other restrictions are imposed on W as well . Most of the problems seem difficult and our results will give just some orientation. The results show that x(1) can be arbitrarily large while the finite subgraphs are very close to bipartite graphs . It is clear from what was said above that this topic is a strange mixture of finite combinatorics and set theory and we recommend it only for those who are interested in both subjects . Finally we want to remind the reader the most striking difference between large chromatic finite and infinite graphs which was discovered by the first two authors about fifteen years ago [4]. While for any k < w there are finite graphs with x(W) > k without any short circuits, [1], a graph with x(W)>K>,w has to contain a complete bipartite graph [k, K + ] for all k <w. Hence such a graph contains all finite bipartite graphs, though it may avoid short odd circuits . Our set-theoretical notation will be standard as for graph theory we will use the notation of our joint paper with F. Galvin [3] with some self-explanatory changes .