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arXiv · 2412.03505

The Zarankiewicz problem on tripartite graphs

Francesco Di Braccio·Freddie Illingworth
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Abstract

In 1975, Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di asked for the smallest $\tau$ such that an $n \times n \times n$ tripartite graph with minimum degree $n + \tau$ must contain $K_{t, t, t}$, conjecturing that $\tau = \mathcal{O}(n^{1/2})$ for $t = 2$. We prove that $\tau = \mathcal{O}(n^{1 - 1/t})$ which confirms their conjecture and is best possible assuming the widely believed conjecture that the Zarankiewicz number satisfies $z(n; t) = \Theta(n^{2 - 1/t})$. Our proof uses a density increment argument. We also construct an infinite family of extremal graphs that are pairwise far apart (requiring the change of $\Omega(n^2)$ edges to get between any two).

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Francesco Di Braccio, Freddie Illingworth. 2024-12-04. The Zarankiewicz problem on tripartite graphs. https://arxiv.org/abs/2412.03505

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