The Ramsey number R(G1β,G2β,G3β) is the smallest positive integer n
such that for all 3-colorings of the edges of Knβ there is a monochromatic
G1β in the first color, G2β in the second color, or G3β in the third
color. We study the bounds on various 3-color Ramsey numbers R(G1β,G2β,G3β), where Giββ{K3β,K3β+e,K4ββe,K4β}. The minimal and maximal
combinations of Giβ's correspond to the classical Ramsey numbers R3β(K3β)
and R3β(K4β), respectively, where R3β(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(K3β,K3β,K4ββe)=17, and by construction we raise the lower bounds on R(K3β,K4ββe,K4ββe) and R(K4β,K4ββe,K4ββe). For some G and H it was known that
R(K3β,G,H)=R(K3β+e,G,H); we prove this is true for several more cases
including R(K3β,K3β,K4ββe)=R(K3β+e,K3β+e,K4ββe).
Ramsey numbers generalize to more colors, such as in the famous 4-color case
of R4β(K3β), where monochromatic triangles are avoided. It is known that 51β€R4β(K3β)β€62. We prove a surprising theorem stating that if
R4β(K3β)=51 then R4β(K3β+e)=52, otherwise R4β(K3β+e)=R4β(K3β).Comment: 12 page