Edge Colourings of Graphs Avoiding Monochromatic Matchings of a Given Size

Let k and ` be positive integers. With a graph G, we associate the quantity ck,`(G), the number of k-colorings of the edge set of G with no monochromatic matching of size `. Consider the function ck,` : N −→ N given by ck,`(n) = max {ck,`(G) : |V (G)| = n}, the maximum of ck,`(G) over all graphs G on n vertices. In this paper, we determine ck,`(n) and the corresponding extremal graphs for all large n and all fixed values of k and `.