The Ramsey Multiplicity of K4

With the help of computer algorithms, we improve the lower bound on the Ramsey multiplicity of K4, and thus show that the exact value of it is equal to 9. The Ramsey multiplicity M(G) of a graph G is defined as the smallest number of monochromatic copies ofG in any two-coloring of edges of KR(G), whereR(G) is the Ramsey number of G, i.e. the smallest integer n such that any two-coloring of edges of Kn contains monochromatic copy of G. The study of Ramsey multiplicity was initiated in 1974 by Harary and Prins [3] who determinedM(G) for all graphsG of order four or less, except forK4 andK4−e. The value of M(K4 − e) was later determined by Schwenk (cited in [2]). The upper bound M(K4) ≤ 12 was given in 1980 by Jacobson [4], and in 1988 Exoo [1] improved it by 3. The only nontrivial lower boundM(K4) ≥ 4 was recently presented by Olpp [7]. In this paper we improve this lower bound and thus show that M(K4) = 9. In the sequel, any two-coloring of the edges of Kn containing k monochromatic copies of K4 is called an (n, k)-coloring. We say that two colorings are isomorphic if the graphs induced by the edges in the first color are isomorphic. Define M(n, k) to be set of all (n, k)-colorings. For a given (n, k)-coloring C let H(C) denote the hypergraph formed by monochromatic copies of K4 in C. Let us defineMd(n, k) to be the subset of all colorings C ∈ M(n, k) such that the maximal vertex degree in H(C) is equal to d. Our computational approach was to generate all nonisomorphic (18, k)-colorings for 4 ≤ k ≤ 8, by iterating an exhaustive enumeration of all possible one vertex extensions of (n − 1, k −m)-colorings to (n, k)-colorings, for m ≥ 0. Let us define E(n−1, k−m,m) to be the subset of all colorings fromM(n, k) which are one vertex extensions of some coloring fromM(n− 1, k −m). ∗ Supported by the State Committee for Scientific Research KBN and academic computer center TASK.