Number of triangles in a multi-partite graph with large minimum degree

A problem on the number of triangles in a multi-partite graph with large minimum degree (proposed by Bollobás, Erdös and Szemerédi [3])

Suppose \( G\) is an \(r \)-partite graph with vertex set consisting of \(r\) parts each of size \(n\). If the minimum degree of \(G\) is at least \(n+t\), is it true that \(G\) always contains \(4t^3\) triangles?

In [3] was shown that such \( G\) contains \( t^3\) triangles, but does not have to contain more than \( 4t^3\) triangles.

Bibliography
1 A. Thomason, A disproof of a conjecture of Erdös in Ramsey theory, J. London Math. Soc. 39 (1989), 246-255.

2 F. Franek and V. Rödl, \( 2\)-colorings of complete graphs with a small number of monochromatic \( K_4\) subgraphs, Discrete Math. 114 (1993), 199-203.

3 B. Bollobás, P. Erdös and E. Szemerédi, On complete subgraphs of \( r\)-chromatic graphs, Discrete Math. 13 (1975), 97-107.