New bounds for some small multicolor Ramsey numbers

The Ramsey number $R(G_1,\dots,G_k)$ is the smallest $n$ such that every $k$-coloring of the edges of $K_n$ contains a monochromatic copy of $G_i$ in color $i$. Ramsey numbers are challenging to compute, and few are known exactly. We use Boolean satisfiability (SAT) solvers to search for structured colorings that give lower bounds, and we show $R(K_4,K_4-e,K_4-e) \ge 35$ and $R(K_3,K_4,C_4,C_4) \ge 49$. Moreover, we tighten some recent upper bounds for multicolor Ramsey numbers for cycles and show $R(C_3,C_6,C_6) = R(C_5,C_6,C_6) = 15$. Finally, we enumerate critical graphs for the numbers $R(C_4,K_{1,s})$ and $R(C_6,K_{1,s})$.