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 $\hat{I}¼>0$ there is an integer $d:=d(S,\hat{I}¼)$ and finite configuration $X\subseteq {\mathbb{R}}^d$ such that any subconfiguration $Y\subseteq X$ with $|Y|\ge \hat{I}¼|X|$ contains a copy of $S$. Complementing this, here we show the existence of $\hat{I}¼:=\hat{I}¼(S)$ 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 $|Z|\ge \hat{I}¼|Y|$ without a copy of $S$.

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