On the maximum number of $r$-cliques in graphs free of complete $r$-partite subgraphs

Abstract

We estimate the maximum possible number of cliques of size $r$ in an $n$-vertex graph free of a fixed complete $r$-partite graph $K_{s_1, s_2, \ldots, s_r}$. By viewing every $r$-clique as a hyperedge, the upper bound on the Turán number of the complete $r$-partite hypergraphs gives the upper bound $O\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. We improve this to $o\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. The main tool in our proof is the graph removal lemma. We also provide several lower bound constructions.

Theorems & Definitions (13)

• Theorem 1
• Proposition 2
• Proposition 3
• Lemma 4
• Lemma 5
• proof
• proof : Proof of \ref{['thm: upper bound']}
• Claim 6
• proof
• proof : Proof of \ref{['thm: lower bound']}