Ramsey numbers upon vertex deletion

TL;DR

This paper disproves a conjecture that deleting a vertex from a graph only slightly changes its Ramsey number, by providing a family of graphs where the change is significantly larger, revealing new complexity in Ramsey theory.

Contribution

The authors construct an infinite family of graphs where vertex deletion causes a super-constant decrease in Ramsey numbers, challenging previous conjectures.

Findings

Deleting a vertex can drastically reduce the Ramsey number for certain graphs.

Existence of graphs with colorings where one color class has negligible density.

Disproof of the conjecture by Conlon, Fox, and Sudakov.

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 $o(1)$.