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arXiv · 2509.23929

Ramsey sequences with bounded clique size

Abhishek Girish Aher·Aparna Lakshmanan S
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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 $\lim_{k \to \infty} \chi(G_k) = \infty$ and $\lim_{k \to \infty} \omega(G_k) \neq \infty$ that is a Ramsey sequence?". In this paper, we solve this problem by demonstrating that the sequence of General shift graphs $Sh(n,k,l)$ 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 $Sh(n,2)$ 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 $E(Sh(n,t-1))$.

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Abhishek Girish Aher, Aparna Lakshmanan S. 2025-09-28. Ramsey sequences with bounded clique size. https://arxiv.org/abs/2509.23929

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