Stability-type results for hereditary properties

The classical Stability Theorem of Erdös and Simonovits can be stated as follows. For a monotone graph property P, let t≥2 be such that t+1=min{_x(H):H∉P }. Then any graph G*ΕP on n vertices, which was obtained by removing at most (1/t+o(1))(n2) edges from the complete graph G=K_n, has edit distance o(n²) to T _n(t), the Turán 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 K_n 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,1/2) 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 T_n(r,s,1/2) almost surely. Here T_n(r,s,1/2) 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 1/2). In addition, several strengthened versions of this result are shown.