  Given a graph $H$ and a function $f:\mathbb{Z}^+ \longrightarrow \mathbb{Z}^+
$, the Ramsey-Tur\'an 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, Erd\H{o}s and
S\'os 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 $RT(n, K_{2r+1}, o(n))$.
