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arXiv · 2209.05455

Ramsey numbers with prescribed rate of growth

Matías Pavez-Signé·Simón Piga·Nicolás Sanhueza-Matamala
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Abstract

Let $R(G)$ be the two-colour Ramsey number of a graph $G$. In this note, we prove that for any non-decreasing function $n \leq f(n) \leq R(K_n)$, there exists a sequence of connected graphs $(G_n)_{n\in\mathbb N}$, with $|V(G_n)| = n$ for all $n \geq 1$, such that $R(G_n) = \Theta(f(n))$. In contrast, we also show that an analogous statement does not hold for hypergraphs of uniformity at least $5$. We also use our techniques to answer a question posed by DeBiasio about the existence of sequences of graphs whose $2$-colour Ramsey number is linear whereas their $3$-colour Ramsey number has superlinear growth.

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Matías Pavez-Signé, Simón Piga, Nicolás Sanhueza-Matamala. 2022-09-12. Ramsey numbers with prescribed rate of growth. https://doi.org/10.37236/11652

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