Ramsey numbers with prescribed rate of growth (Q6199183)

Summary: Let \(R(G)\) be the 2-colour Ramsey number of a graph \(G\). In this note, we prove that for any non-decreasing function \(n \leqslant f(n) \leqslant R(K_n)\), there exists a sequence of connected graphs \((G_n)_{n\in\mathbb{N}}\), with \(|V(G_n)| = n\) for all \(n \geqslant 1\), 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 in the affirmative a question posed by \textit{L. DeBiasio} [``Graphs with linear Ramsey number for two colors, but super-linear Ramsey number for three colors?'', \url{https://mathoverflow.net/q/385498}] about the existence of sequences of graphs, but whose 2-colour Ramsey number is linear whereas their 3-colour Ramsey number has superlinear growth.