The Zarankiewicz problem on tripartite graphs

TL;DR

This paper proves a conjecture about the minimum degree condition in tripartite graphs that guarantees the presence of a complete tripartite subgraph, confirming a long-standing open problem in extremal graph theory.

Contribution

It establishes the optimal bound for the minimum degree condition in tripartite graphs to contain a $K_{t,t,t}$, confirming a conjecture and advancing understanding of the Zarankiewicz problem.

Findings

Proves $ au = ext{O}(n^{1 - 1/t})$ for tripartite graphs.

Confirms the conjecture that $ au = ext{O}(n^{1/2})$ for $t=2$.

Constructs extremal graphs matching the bounds.

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 $\operatorname{ex}(n, K_{t, t}) = \Theta(n^{2 - 1/t})$. Our proof uses a density increment argument. We also construct an infinite family of extremal graphs.