ON SOME MULTICOLOR RAMSEY NUMBERS INVOLVING K3+e AND K4-e

The Ramsey number R(G(1), G(2), G(3)) is the smallest positive integer n such that for all 3-colorings of the edges of K-n there is a monochromatic G(1) in the first color, G(2) in the second color, or G(3) in the third color. We study the bounds on various 3-color Ramsey numbers R(G(1), G(2), G(3)), where G(i) is an element of {K-3, K3+e, K-4 - e, K-4}. The minimal and maximal combinations of G(i)'s correspond to the classical Ramsey numbers R-3(K-3) and R-3(K-4), respectively, where R-3(G) = R(G, G, G). Here, we focus on the much less studied combinations between these two cases. Through computational and theoretical means we establish that R(K-3, K-3, K-4 - e) = 17, and by construction we raise the lower bounds on R(K-3, K-4 - e, K-4 - e) and R(K-4, K-4 - e, K-4 - e). For some G and H it was known that R(K-3, G, H) = R(K-3 + e, G, H); we prove this is true for several more cases including R(K-3, K-3, K-4 - e) = R(K-3 + e, K-3 + e, K-4 - e). Ramsey numbers generalize to more colors, such as in the famous 4-color case of R-4(K-3), where monochromatic triangles are avoided. It is known that 51 <= R-4(K-3) <= 62. We prove a surprising theorem stating that if R-4(K-3) = 51, then R-4(K-3 + e) = 52, otherwise R-4(K-3 + e) = R-4(K-3).