Stability-type results for hereditary properties

The classical Stability Theorem of Erdos and Simonovits can be stated as follows. For a monotone graph property P, let t≥2 be such that t+1=minlχ(H):H∉∈Pr. Then any graph G*∈P on n vertices, which was obtained by removing at most $(1/t+o (1)) \left(_{2}^{n}\right)$ edges from the complete graph G=Kn, has edit distance o(n2) to Tn(t), the Turan graph on n vertices with t parts. In this paper we extend the above notion of stability to hereditary graph properties. It turns out that to do so the complete graph Kn has to be replaced by a random graph. For a hereditary graph property P, consider modifying the edges of a random graph G=G(n,½) to obtain a graph G* that satisfies P in (essentially) the most economical way. We obtain necessary and sufficient conditions on P, which guarantee that G* has a unique structure. In such cases, for a pair of integers (r, s), which depends on P, G*, has distance o(n2) to a graph Tn(r, s, ½) almost surely. Here Tn(r, s, ½) denotes a graph, which consists of almost equal-sized r+s parts, r of them induce an independent set, s induce a clique and all the bipartite graphs between parts are quasi-random (with edge density ½. In addition, several strengthened versions of this result are shown. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 65–83, 2009