The Turan number of Berge hypergraphs with stable properties

Let G be a graph on n vertices, P be a property defined on all graphs on n vertices and k be a positive integer. A property P is said to be k-stable, if whenever G + uv has the property P and the sum of the degrees of u and v in G is at least k, then G itself has the property P. Assume property P is k-stable, and G is an n-vertex graph with minimum degree at least d and without the property P. In this paper we obtain the maximum possible number of r-cliques in the graph G. Furthermore, assume the property P of containing a graph in a family .F is k-stable, we determine the Turan number ex(r)(n, Berge -.F) for the case r <= [ (k-1) (2) ] -1 and characterize the extremal hypergraphs. For the case [(k-1) (2) ] <= r <= k, we give an upper bound on ex(r)(n, Berge -.F). Several known results are generalized.(c) 2023 Elsevier B.V. All rights reserved.