paper

Diagonal Ramsey numbers for wheels

arXiv:2605.22116

Abstract

The Ramsey number $R (G_{1}, G_{2})$ is the smallest integer $N$ such that any red-blue coloring of the edges of the complete graph $K_{N}$ contains either a red copy of $G_{1}$ or a blue copy of $G_{2}$ . In 2022, the third author and others gave lower and upper bounds of the Ramsey number $R (W_{n}, W_{n})$ , where $W_{n}$ is the wheel graph with $n$ vertices. In this paper, we improve their bounds by showing that $3 n - 2 \leq R (W_{n}, W_{n}) \leq 6 n - 6$ for even $n \geq 8$ and $2 n \leq R (W_{n}, W_{n}) \leq \frac{9 n - 7}{2}$ for odd $n \geq 7$ . Furthermore, we give recursive bounds for the $k$ -colored Ramsey number for $W_{n}$ .