A conjecture of Goodman and the multiplicities of graphs

We prove a about the maximum number of monochromatic VLJ. • kU"."OVU T,WO-(;OllC)nng of the of Kn with fixed number of Moreover, this result is used for the multiplicities M(G; n) of some small where M(G; n) defined as the smallest number of G in any two-coloring of the of Kn-this the determination of with at most three edges. 1. In the we consider of the of the complete K n , for short colorings, and use red and blue as our colors. For a graph G and coloring C of K n) we denote the number of monochromatic copies of G in C by N c (G) (or just N(G), if it is clear which coloring is referred to). The m1.dtiplicity M(Gj n) of a graph G and a positive n is defined as ffiJn over all colorings C of Kn. It includes the Ramsey number r(G), which is the smallest n such that M(Gj n) is positive, and the Ramsey multiplicity R(G), which is M(Gjr(G)). Those colorings C in which M(G;n) attained are called minimizing colorings. There are very few exact results about multiplicity: The only graphs G for which M(G; n) was known for all n E IN were the triangle K3 (Goodman [3]), the path and the stars K1,m for all m E IN (Czerniakiewicz [2], Burr and Rosta [1 J). In Section 3, we will determine the multiplicities of 3K 2 , and P3 U e, that we have the exact values of n) for all graphs G with at most three Goodman also determined min Nc(K3) where the minimum is taken over the c colorings C of with some fixed number of red and blue edges. Moreover, he made a conjecture about max Nc(K3) with the same constraint. (Without this constraint c the maximum is trivially attained in a coloring where all edges have the same color.) This conjecture will be proved in Section 2.