Turán Numbers of Expanded Intersecting Cliques in 3-graphs

Abstract

Let $\ell >r\geqslant 3$. Given a 2-graph F, the expansion $F^{(r)}$ of F is an r-graph obtained from F by adding $r-2$ new vertices into each edge. When F is a clique of order $\ell$, the Turán number $\textrm{ex}(n, F^{(r)})$ was first asymptotically determined by Mubayi (J Comb Theory Ser B 96:122–134, 2006) and exactly computed by Pikhurko (J Comb Theory Ser B 103:220–225, 2013). Let $F_{k,\ell }$ be the 2-graph on $(\ell -1)k+1$ vertices consisting of k cliques of order $\ell$ intersecting at exactly one vertex. We determine the exact Turán number $\textrm{ex}(n, F_{k,\ell }^{(3)})$ for all $\ell >3$, $k\geqslant 1$ and sufficiently large n, as well as the corresponding extremal graphs.