The Colouring Number

Abstract

It was proved by Tutte in [23] that for every finite n an n -chromatic triangle-free graph exists. Later Erdos and Rado [10] extended this result to infinite n. Another extension of a different kind was found by Erdos in [5]: if n, s are finite numbers, there exists an n-chromatic graph without circuits of length at most s. The proof was non-constructive, a construction was later found by Lov£sz [16] (see also [11,18,19]). The obvious common generalization of these results is, however, false. In fact, the following theorem of Erdos and Hajnal, discovered in [7], holds: every graph G of uncountable chromatic number contains a circuit of length 4. Even more is true, namely, that for every finite n, the complete bipartite graph on n, Kx vertices is contained in G. They also proved that those graphs must contain infinite paths, and gave examples with arbitrarily large chromatic number not containing odd circuits of length 3, 5, ...,2s + 1, where s is a given finite number. Concerning odd circuits, it was proved by Erdos, Hajnal, and Shelah in [8] that if the chromatic number of G is uncountable, then there exists an s0 such that for every s^s0, G contains a circuit of length 2s + 1; s0, of course, depends on G. Another obligatory—in the above sense—graph was exhibited by A. Hajnal: the vertices are {xh yt i co and it contains all