Abstract
Given a hereditary family G of admissible graphs and a function λ(G) that linearly depends on the statistics of order-κ subgraphs in a graph G, we consider the extremal problem of determining λ(n,G), the maximum of λ(G) over all admissible graphs G of order n. We call the problem perfectly B-stable for a graph B if there is a constant C such that every admissible graph G of order n⩾C can be made into a blow-up of B by changing at most C(λ(n,G)−λ(G))(n2) adjacencies. As special cases, this property describes all almost extremal graphs of order n within o(n2) edges and shows that every extremal graph of order n⩾C is a blow-up of B.We develop general methods for establishing stability-type results from flag algebra computations and apply them to concrete examples. In fact, one of our sufficient conditions for perfect stability is stated in a way that allows automatic verification by a computer. This gives a unifying way to obtain computer-assisted proofs of many new results.
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