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arXiv · 1910.11048

Turán number of bipartite graphs with no $K_{t,t}$

Benny Sudakov·István Tomon
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Abstract

The extremal number of a graph $H$, denoted by $\mbox{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices that does not contain $H$. The celebrated Kővári-Sós-Turán theorem says that for a complete bipartite graph with parts of size $t\leq s$ the extremal number is $\mbox{ex}(K_{s,t})=O(n^{2-1/t})$. It is also known that this bound is sharp if $s>(t-1)!$. In this paper, we prove that if $H$ is a bipartite graph such that all vertices in one of its parts have degree at most $t$, but $H$ contains no copy of $K_{t,t}$, then $\mbox{ex}(n,H)=o(n^{2-1/t})$. This verifies a conjecture of Conlon, Janzer and Lee.

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Benny Sudakov, István Tomon. 2019-10-24. Turán number of bipartite graphs with no $K_{t,t}$. https://arxiv.org/abs/1910.11048

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