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Abstract.The structure of labeled A"/+1-free graphs is investigated asymptotically. Through a series of stages of successive refinement the structure of "almost all" such graphs is found sufficiently precisely to prove that they are in fact /-colorable (/-partite). With the asymptotic information obtained it is shown also that in the class of A^/+1-free graphs there is a first-order labeled 0-1 law. With this result, and those cases already known, we can say that any infinite class of finite undirected graphs with amalgamations, induced subgraphs and isomorphisms has a 0-1 law.Introduction. In this paper we investigate the asymptotic behavior of A^/+1-free graphs, i.e., of finite undirected graphs which do not contain as a subgraph the complete graph Kl+l on / 4-1 vertices. In the first part asymptotic results about the number and the structure of labeled AT/+1-free graphs are obtained. These results are applied in the second part in order to study the labeled asymptotic probabilities of first-order sentences on the class £f{l) of all A^/+1-free graphs. We now describe briefly the main theorems of this paper.Let / > 2. If G is an /-colorable graph, then obviously G cannot contain as a subgraph the complete graph Kl+l on / + 1 vertices. But it is well known (see e.g. Bollobas [1980]) that there are A^/+rfree graphs with arbitrarily large chromatic number. Hence, for n large enough, the number of /C/+1-free graphs on {1,...,«} is strictly greater than the number of /-colorable graphs. In contrast to this we show that "almost all" Kl+l-free graphs are /-colorable. More precisely, we establish in the first part of this paper:

In this note we announce some results about the asymptotic behavior of K m -free graphs. These are the undirected finite graphs which do not contain a complete graph K m with m vertices (an m-clique) as a subgraph. It is obvious that every graph which contains a clique of size / + 1 is not /-colorable, and hence has chromatic number at least I -f 1. Also it is well known that there are if/+i-free graphs of arbitrarily large chromatic number. In contrast to this we show that "almost-all" Zf/+i-free graphs are /-colorable, for any / > 2. More precisely, we establish In addition to the asymptotic enumeration given by Theorem 1, we derive detailed information about the structure of almost all ifj+i-free graphs. We use this to prove that the labeled asymptotic probability of any first-order property on the class S (I) of all finite Ki+i-free graphs is either 0 or 1. C. W.