Lower Bounds on the Number of Monochromatic Matchings in $K_{2s+t-1}$

Abstract

Cockayne and Lorimer proved that in any 2-edge-coloring of$K_{2s+t-1}$ $(s\ge t\ge 1)$ there exists an s-matching receiving the same color (we call such matching monochromatic matching) or a t-matching having another color. In this paper, we show that the number of such monochromatic matchings may be exponentially many. We prove that in any 2-edge-coloring of $K_{2s+t-1} (s \ge t \ge 1)$ there are either at least $2^{[t/2]}f(n)$ many monochromatic s-matchings or at least $2^{[t/2]}$ many t-matchings of another color, where $f(n)=(2n+1)!!$, and $n=s-t$. These generalize a Ramsey number by Erdös, Gyárfás and Pyber, Gerencsér and Gyárfás and improve a lower bound of the number of such matching by Cao and Ren as well.