Ramsey sequences with bounded clique size

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 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 $\lim_{k \to \infty} \chi(G_k) = \infty$ and $\lim_{k \to \infty} \omega(G_k) \neq \infty$ that is a Ramsey sequence?". In this paper, we solve this problem by constructing a Ramsey sequence $\{G_k\}$ with a bounded clique number such that $\lim_{k \to \infty} \chi(G_k) = \infty$. Furthermore, using the observation that any monotonic increasing sequence of graphs that contains a Ramsey sequence as a subgraph is also Ramsey, we can generate infinitely many Ramsey sequences using this example.