Nowhere dense Ramsey sets

Abstract

A set of points $S$ in Euclidean space $\mathbb{R}^d$ is called \textit{Ramsey} if any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the area, a stronger ``density'' concept was considered in [J. Amer. Math. Soc. 3, 1--7, 1990]: If $S$ is a $d$-dimensional simplex, then for any $\mu>0$ there is an integer $d:=d(S,\mu)$ and finite configuration $X\subseteq \mathbb{R}^d$ such that any subconfiguration $Y\subseteq X$ with $|Y|\geq \mu |X|$ contains a copy of $S$. Complementing this, here we show the existence of $\mu:=\mu(S)$ and of an infinite configuration $X\subseteq \mathbb{R}^{\infty}$ with the property that any finite coloring of $X$ yields a monochromatic copy of $S$, yet for any finite set of points $Y\subseteq X$ contains a subset $Z\subseteq Y$ of size $|Z|\geq \mu |Y|$ without a copy of $S$.