The Structure of Hereditary Properties and Colourings of Random Graphs

in the probability space ? Second, does there exist a constant such that the -chromatic number of the random graph is almost surely ? The second question was posed by Scheinerman (SIAM J. Discrete Math. 5 (1992) 74–80).

The two questions are closely related and, in the case p=1/2, have already been answered. Prömel and Steger (Contemporary Mathematics 147, Amer. Math. Soc., Providence, 1993, pp. 167-178), Alekseev (Discrete Math. Appl. 3 (1993) 191-199) and the authors ( Algorithms and Combinatorics 14 Springer-Verlag (1997) 70–78) provided an approximation which was used by the authors (Random Structures and Algorithms 6 (1995) 353–356) to answer the -chromatic question for p=1/2. However, the approximating properties that work well for p=1/2 fail completely for .

In this paper we describe a class of properties that do approximate in , in the following sense: for any desired accuracy of approximation, there is a property in our class that approximates to this level of accuracy. As may be expected, our class includes the simple properties used in the case p=1/2.

The main difficulty in answering the second of our two questions, that concerning the -chromatic number of , is that the number of small -graphs in has, in general, large variance. The variance is smaller if we replace by a simple approximation, but it is still not small enough. We overcome this by considering instead a very rigid non-hereditary subproperty of the approximating property; the variance of the number of small -graphs is small enough for our purpose, and the structure of is sufficiently restricted to enable us to show this by a fine analysis.