Ramsey numbers upon vertex deletion

TL;DR

This work disproves the Conlon–Fox–Sudakov conjecture by constructing explicit $(n{+}1)$-vertex graphs $G_{k,n}$ and $H_{k,n}$ with $r(H_{k,n})=n$ and $r(G_{k,n})>nk$, yielding a super-constant gap after deleting a single vertex. It also proves a universal upper bound $r(G) \le C\sqrt{n\log n}\, r(H)$ and develops the framework of $\varepsilon$-Ramsey-balanced graphs, for which the conjecture holds, while showing that not all graphs are so balanced, implying Ramsey colorings with a color-class density $o(1)$. In the multicolor setting ($q\ge 3$), they obtain a polynomially larger lower bound $r(G;q) > n^{1+\alpha}$ with $\alpha=\frac{3q-5}{8q\log q}-o(1)$, illustrating a stronger gap. The paper thus clarifies when vertex deletion can cause large changes in Ramsey numbers, linking degeneracy and balance properties, and raises open problems about degeneracy thresholds, average-case behavior, and edge-deletion analogues.

Abstract

Given a graph $G$, its Ramsey number $r(G)$ is the minimum $N$ so that every two-coloring of $E(K_N)$ contains a monochromatic copy of $G$. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from $G$, the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super-constant factor. One consequence of this result is the following. There exists a family of graphs $\{G_n\}$ so that in any Ramsey coloring for $G_n$ (that is, a coloring of a clique on $r(G_n)-1$ vertices with no monochromatic copy of $G_n$), one of the color classes has density $o(1)$.

Theorems & Definitions (27)

• Conjecture 1.1: MR4170438
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Lemma 2.1
• Lemma 2.2
• proof : Proof of \ref{['lem:H-ub']}
• proof : Proof of \ref{['lem:G-lb']}
• Proposition 3.1
• proof : Proof of \ref{['thm:sqrt-ub']}