A new bound for the 2/3 conjecture

Abstract

We show that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of the neighbors of these vertices in one of the colors is at least 2n/3. The previous best value, proved by Erdos, Faudree, Gould, Gy\'arf\'as, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.