On subgraphs of tripartite graphs

Abstract

Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di [Discrete Math 13 (1975), 97--107] investigated a tripartite generalization of the Zarankiewicz problem: what minimum degree forces a tripartite graph with $n$ vertices in each part to contain an octahedral graph $K_3(2)$? They proved that $n+2^{-1/2}n^{3/4}$ suffices and suggested it could be weakened to $n+cn^{1/2}$ for some constant $c>0$. In this note we show that their method only gives $n+ (1+o(1)) n^{11/12}$ and provide many constructions that show if true, $n+ c n^{1/2}$ is better possible.