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On the maximum number of triangles in tripartite graphs with no $4$-cycles between any two parts
Let $G$ be a $3$-partite graph with $k$ vertices in each part such that the bipartite graph induced by any two parts contains no cycle of length four. Fischer and Matoušek [J. Combin. Theory Ser. A, 2001] asked for the maximum number of triangles in such a graph. They obtained the lower bound $(1-o(
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