New bounds for some small multicolor Ramsey numbers

William J. Wesley

paper

New bounds for some small multicolor Ramsey numbers

arXiv:2509.03784

Abstract

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 $R (K_{4}, K_{4} - e, K_{4} - e) \geq 35$ and $R (K_{3}, K_{4}, C_{4}, C_{4}) \geq 49$ . 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 $R (C_{6}, K_{1, s})$ .

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