Remarks on an Edge-coloring Problem

We consider a multicolored version of a problem that was originally proposed by Erdős and Rothschild. For positive integers n and r, we look for n-vertex graphs that admit the maximum number of r-edge-colorings with no copy of a triangle where exactly two colors appear. It turns out that for 2 ≤ r ≤ 12 colors and n sufficiently large, the complete bipartite graph on n vertices with balanced bipartition (the n-vertex Turán graph for the triangle) yields the largest number of such colorings, and this graph is unique with this property.