Complete tripartite subgraphs of balanced tripartite graphs with large minimum degree

In 1975 Bollobás, Erdős, and Szemerédi asked what minimum degree guarantees an octahedral subgraph K₃(2) in any tripartite graph G with n vertices in each vertex class. We show that δ(G)≥ n+2n5/6 suffices thus improving the bound n+(1+o(1))n11/12 of Bhalkikar and Zhao obtained by following their approach. Bollobás, Erdős, and Szemerédi conjectured that n+cn1/2 suffices and there are many K₃(2)-free tripartite graphs G with δ(G)≥ n+cn1/2. We confirm this conjecture under the additional assumption that every vertex in G is adjacent to at least (1/5+ε)n vertices in any other vertex class.