Planar polynomials and an extremal problem of Fischer and Matoušek

Let G be a 3-partite graph with k vertices in each part and suppose that between any two parts, there is no cycle of length four. Fischer and Matoušek asked for the maximum number of triangles in such a graph. A simple construction involving arbitrary projective planes shows that there is such a graph with (1−o(1))k3/2 triangles, and a double counting argument shows that one cannot have more than (1+o(1))k7/4 triangles. Using affine planes defined by specific planar polynomials over finite fields, we improve the lower bound to (1−o(1))k5/3.