Complete solution for the rainbow numbers of matchings

For a given graph HH and a positive nn, the rainbow number of  HH, denoted by rb(n,H)rb(n,H), is the minimum integer kk so that in any edge-coloring of KnKn with kk colors there is a copy of HH whose edges have distinct colors. In 2004, Schiermeyer determined rb(n,kK2)rb(n,kK2) for all n≥3k+3n≥3k+3. The case for smaller values of nn (namely, n∈[2k,3k+2]) remained generally open. In this paper we extend Schiermeyer’s result to all plausible nn and hence determine the rainbow number of matchings.