Bounds on the Threshold Ramsey Multiplicity of Ramsey Numbers with Many Colors

The Ramsey number R(s,t) is the least integer n such that any coloring of the edges of Kn with two colors produces either a monochromatic Ks in one color or a monochromatic Kt in the other. If s=t, we call R(s,s) a diagonal Ramsey number. The threshold Ramsey multiplicity of a diagonal Ramsey number R(s,s), denoted by m(s,s) or m2(s), is the smallest number of copies of a monochromatic Ks that can be found in any coloring of the edges of KR(s,s). For instance, m2(2)=1, m2(3)=2, and m2(4)=9. We derive upper bounds for multicolor, off-diagonal threshold Ramsey multiplicities. In the diagonal two-color case, the resulting explicit numerical bounds improve those obtained from the elementary random-coloring estimate for 5≤s≤8. In the multicolor case, we recover the known value m(3,3,3)=5 and obtain the bound m(3,3,4)≤8. We conclude with a general framework for seeking further improvements.