Ramsey numbers upon vertex deletion

Abstract

Given a graph $G$, its Ramsey number $r\left(G\right)$ is the minimum $N$ so that every two-coloring of $E\left(K_N\right)$ 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 $\left\{G_n\right\}$ so that in any Ramsey coloring for $G_n$ (i.e., a coloring of a clique on $r\left(G_n\right)-1$ vertices with no monochromatic copy of $G_n$), one of the color classes has density $o\left(1\right)$.