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

In this paper graphs with uncountable chromatic numbers will be studied. As usual, a graph is an ordered pair G = ( V , E), where V is an arbitrary set (the set of vertices), E is a set of unordered pairs from V (the set of edges). A function : V + x (n a cardinal), is a good coloring of G if and only if f ( x ) ¢ f ( y ) whenever x and y are jo ined i.e. joined vertices get different colors. The chromatic number of G, Chr (G) is the minimal cardinal n onto which a good coloring of G exists. The following statement was proved by Tutte, Zykov, Ung~r, Mycielski and possibly by many other people (see e.g. [9], [11]): for every finite n there exists an n-chromatic triangle-free graph. P. Erd6s and R. Rado proved in [5] that for every cardinality g there exists a triangle-free n-chromatic glaph of cardinality n. P. Erd6s proved (see [1]) that for any given finite n and s there exist n-chromatic graphs without circuits of length ~s . His proof was non-constructive. A constructive p~oof was later found by L. Lov~isz [8]. However quite surprisingly, the natural common generalization of these theorems turned out to be false: if a graph has chromatic number ~ R1 then it necessarily contains a four-cycle, moreover every finite bipartite graph must be contained in such a graph. As examples show, for every s<~o and arbitrary cardinality n there exist graphs with chromatic number n (and of cardinality n) without any odd circuit of length ~ s (see [3]). Thus, all finite obligatory graphs are described. The next problems are the following: find the obligatory classes of finite graphs and the obligatory countable graphs. Both problems seem to be very difficult (see [4]). In this paper we try to attack the second problem by displaying two graphs F and A which are very close together and though F is obligatory A is not (at least consistently).