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

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.