Ramsey simplicity of random graphs
Simona Boyadzhiyska, Dennis Clemens, Shagnik Das, Pranshu Gupta
TL;DR
The paper investigates the q-Ramsey simplicity of the random graph G(n,p), introducing the threshold tilde{q}(H) governing when H is q-Ramsey simple for growing numbers of colours. It develops a transference framework linking the neighbourhood of the minimum-degree vertex to global Ramsey behavior and uses explicit geometric constructions to realize simplicity in sparse regimes, while proving non-simplicity and providing sharp bounds in intermediate and dense regimes through the analysis of the subgraph F=H[N(u)]. The results reveal a non-monotone, regime-dependent landscape where tilde{q}(H) can range from infinity to 1, and even exhibit finite thresholds in intermediate densities, with abundance phenomena showing that minimal Ramsey graphs can have arbitrarily many vertices achieving the minimum degree. These findings advance understanding of Ramsey properties in random hosts and connect combinatorial constructions, probabilistic tools, and Ramsey theory in a unified framework with potential implications for related extremal problems.
Abstract
A graph $G$ is $q$-Ramsey for another graph $H$ if in any $q$-edge-colouring of $G$ there is a monochromatic copy of $H$, and the classic Ramsey problem asks for the minimum number of vertices in such a graph. This was broadened in the seminal work of Burr, Erdős, and Lovász to the investigation of other extremal parameters of Ramsey graphs, including the minimum degree. It is not hard to see that if $G$ is minimally $q$-Ramsey for $H$ we must have $δ(G) \ge q(δ(H) - 1) + 1$, and we say that a graph $H$ is $q$-Ramsey simple if this bound can be attained. Grinshpun showed that this is typical of rather sparse graphs, proving that the random graph $G(n,p)$ is almost surely $2$-Ramsey simple when $\frac{\log n}{n} \ll p \ll n^{-2/3}$. In this paper, we explore this question further, asking for which pairs $p = p(n)$ and $q = q(n,p)$ we can expect $G(n,p)$ to be $q$-Ramsey simple. We resolve the problem for a wide range of values of $p$ and $q$; in particular, we uncover some interesting behaviour when $n^{-2/3} \ll p \ll n^{-1/2}$.