The Three-Dimensional Erdős Box Problem Has Exponent $11/4$

Abstract

The Zarankiewicz problem for $3$-uniform hypergraphs asks for the maximum number $z(n)$ of edges in a tripartite hypergraph with $n$ vertices in each part containing no copy of $K_{2,2,2}^{(3)}$ (a ``box''). Erdős (1964) proved $z(n) = O(n^{11/4})$. The best previously known lower bound was $Ω(n^{8/3})$, due to Katz, Krop, and Maggioni (2002). We construct a family of box-free hypergraphs matching Erdős's upper bound: for each $q = 2^m$ ($m \geq 1$), our hypergraph has $q^4$ vertices in each part and $q^{11}$ edges, establishing that $z(n) = Θ(n^{11/4})$. The construction is algebraic, defined over ${\bf F}_{q^3}$ via the power map $τ(s) = s^{q^2-q+1}$. The proof shows that the direction-$1$ finite differences of $τ$ partition ${\bf F}_{q^3}$ into pairwise skew affine lines over ${\bf F}_q$, preventing boxes from forming.