On subgraphs of tripartite graphs

Bollobás, Erdős, and Szemerédi [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 $n+2^{-1\mathrm{/}2}{n}^{3\mathrm{/}4}$ suffices and suggested it could be weakened to $n+c{n}^{1\mathrm{/}2}$ for some constant $c>0$. In this note we show that their method only gives $n+(1+o(1))n^{11\mathrm{/}12}$ and provide many constructions that show if true, $n+c{n}^{1\mathrm{/}2}$ is better possible.