Ramsey numbers with prescribed rate of growth

TL;DR

This paper constructs sequences of graphs with prescribed growth rates of their Ramsey numbers, demonstrating the flexibility of Ramsey number growth in graphs but not in higher uniform hypergraphs, and addressing a question about multi-color Ramsey numbers.

Contribution

It proves the existence of graph sequences with Ramsey numbers growing at any prescribed rate within bounds, and shows such a result does not extend to hypergraphs of uniformity at least 5.

Findings

Constructed graph sequences with Ramsey numbers matching any non-decreasing function up to R(K_n)

Demonstrated the non-existence of similar sequences for certain hypergraphs

Answered a question about differing growth rates of 2-color and 3-color Ramsey numbers

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.