Abstract: Let $\ell$ > r ≥ 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 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 ex(n, F_k,$\ell$⁽³⁾) for all $\ell$ > 3, k ≥ 1 and sufficiently large n, as well as the corresponding extremal graphs.

Key words: Expansion, Turán number, 3-Graph, Intersecting cliques, Stability