An extension of Turán's theorem, uniqueness and stability (Q463041)

Summary: We determine the maximum number of edges of an \(n\)-vertex graph \(G\) with the property that none of its \(r\)-cliques intersects a fixed set \(M\subset V(G)\). For \((r-1)|M|\geqslant n\), the \((r-1)\)-partite Turán graph turns out to be the unique extremal graph. For \((r-1)|M|<n\), there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.