On Some Multicolor Ramsey Numbers Involving $K_3+e$ and $K_4-e$

The Ramsey number R(G1, G2, G3) is the smallest positive integer n such that for all 3-colorings of the edges of Kn there is a monochromatic G1 in the first color, G2 in the second color, or G3 in the third color. We study the bounds on various 3-color Ramsey numbers R(G1, G2, G3), where Gi ∈ {K3,K3 + e,K4 − e,K4}. The minimal and maximal combinations of Gi’s correspond to the classical Ramsey numbers R3(K3) and R3(K4), respectively, where R3(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(K3,K3,K4 − e) = 17, and by construction we raise the lower bounds on R(K3,K4− e,K4− e) and R(K4,K4− e,K4− e). For some G and H it was known that R(K3, G,H) = R(K3 + e,G,H); we prove this is true for several more cases including R(K3,K3,K4−e) = R(K3+e,K3+e,K4−e). Ramsey numbers generalize to more colors, such as in the famous 4-color case of R4(K3), where monochromatic triangles are avoided. It is known that 51 ≤ R4(K3) ≤ 62. We prove a surprising theorem stating that if R4(K3) = 51 then R4(K3 + e) = 52, otherwise R4(K3 + e) = R4(K3).