Edge-colorings avoiding fixed rainbow stars

We consider an extremal problem motivated by a question of Erdős and Rothschild and by a paper of Balogh, who considered edge-colorings of graphs avoiding fixed subgraphs with a prescribed coloring. Given r≥t≥2, we look for n-vertex graphs that admit the maximum number of r-edge-colorings such that at most t−1 colors appear in edges incident with each vertex. For large n, we show that, with the exception of the case t=2, the complete graph Kn is always the unique extremal graph. We also consider generalizations of this problem.