Stability-type results for hereditary properties

The classical Stability Theorem of Erdős and Simonovits can be stated as follows. For a monotone graph property P, let t ≥ 2 be such that t + 1 = min{χ(H) : H / ∈ P}. Then any graph G∗ ∈ P on n vertices, which was obtained by removing at most (1t + o(1)) ( n 2 ) edges from the complete graph G = Kn, has edit distance o(n) to Tn(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 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, 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(n) to a graph Tn(r, s, 1 2 ) almost surely. Here Tn(r, s, 12 ) 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 12 ). In addition, several strengthened versions of this result are shown.