Ramsey-Turán numbers for intersecting odd cliques

Abstract

Given a graph $H$ and a function $f:{\mathbb{Z}}^{+}⟶{\mathbb{Z}}^{+}$, the Ramsey-Turán number of $H$ and $f$, denoted by $RT(n,H,f(n))$, is the maximum number of edges a graph $G$ on $n$ vertices can have, which does not contain $H$ as a subgraph and also does not contain a set of $f(n)$ independent vertices. Let $r$ be a positive integer. In 1969, Erdős and Sós proved that $RT(n,K_{2r+1},o(n))=\frac{n^2}{2}(1-\frac{1}{r})+o(n^2)$. Let $F_k(2r+1)$ denote the graph consisting of $k$ copies of complete graphs $K_{2r+1}$ sharing exactly one vertex. In this paper, we show that $RT(n,F_k(2r+1),o(n))=\frac{n^2}{2}(1-\frac{1}{r})+o(n^2)$, which is of the same magnitude with $RT(n,K_{2r+1},o(n))$.