Size of Monochromatic Double Stars in Edge Colorings

摘要

We show that in every r-coloring of the edges of K n there is a monochromatic double star with at least $frac{n(r+1)+r-1}{r^2}$ vertices. This result is sharp in asymptotic for r = 2 and for r≥ 3 improves a bound of Mubayi for the largest monochromatic subgraph of diameter at most three. When r-colorings are replaced by local r-colorings, our bound is $frac{n(r+1)+r-1}{r^2+1}$ .