On the Tur\'{a}n number of the expansion of the $t$-fan

Abstract

The $t$-fan is the graph on $2t+1$ vertices consisting of $t$ triangles which intersect at exactly one common vertex. For a given graph $F$, the $r$-expansion $F^r$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by adding $r-2$ distinct new vertices to each edge of $F$. We determine the Tur\'an number of the 3-expansion of the $t$-fan for sufficiently large $n$.