Random multilinear maps and the Erd\H{o}s box problem

TL;DR

This paper introduces a novel approach using random multilinear maps to establish improved lower bounds for the Erdős box problem, advancing understanding of extremal hypergraph configurations.

Contribution

It presents new lower bounds for the Erdős box problem by applying random multilinear maps, improving upon previous results by Gunderson, R"{o}dl, and Sidorenko.

Findings

Established new lower bounds for the Erdős box problem.

Enhanced understanding of extremal numbers in hypergraphs.

Improved upon previous bounds by notable researchers.

Abstract

By using random multilinear maps, we provide new lower bounds for the Erd\H{o}s box problem, the problem of estimating the extremal number of the complete $d$-partite $d$-uniform hypergraph with two vertices in each part, thereby improving on work of Gunderson, R\"{o}dl and Sidorenko.