Ramsey sequences with bounded clique size

Abhishek Girish Aher; Aparna Lakshmanan S

paper

Ramsey sequences with bounded clique size

arXiv:2509.23929

Abstract

A sequence of graphs ${G_{k}}$ is a Ramsey sequence if for every positive integer $k$ , the graph $G_{k}$ is a proper subgraph of $G_{k + 1}$ , and there exists an integer $n > k$ such that every red-blue edge coloring of $G_{n}$ contains a monochromatic copy of $G_{k}$ . Among the wide range of open problems in Ramsey theory, an interesting open question is ``Does there exist an ascending sequence ${G_{k}}$ with $lim_{k \to \infty} χ (G_{k}) = \infty$ and $lim_{k \to \infty} ω (G_{k}) \neq = \infty$ that is a Ramsey sequence?". In this paper, we solve this problem by demonstrating that the sequence of General shift graphs $S h (n, k, l)$ is a Ramsey sequence which satisfies the conditions given in the open problem. We further present an alternative proof of the generalised Ramsey theorem by establishing explicitly that $S h (n, 2)$ is a Ramsey sequence for $k$ -edge-colorings, and by observing the natural bijection between the set of all $t$ -element subsets of $[n]$ and the edge set $E (S h (n, t - 1))$ .