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arXiv · 2402.16818

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

József Balogh·Suyun Jiang·Haoran Luo
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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 Tur\'an 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.

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József Balogh, Suyun Jiang, Haoran Luo. 2024-02-26. On the maximum number of $r$-cliques in graphs free of complete $r$-partite subgraphs. https://arxiv.org/abs/2402.16818

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