Nowhere dense Ramsey sets

A set of points S in Euclidean space Rᵈ is called Ramsey if any finite partition of R^ 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 >0 there is an integer d: =d (S, ) and finite configuration X Rᵈ such that any subconfiguration Y X with |Y| |X| contains a copy of S. Complementing this, here we show the existence of: = (S) and of an infinite configuration X R^ with the property that any finite coloring of X yields a monochromatic copy of S, yet for any finite set of points Y X contains a subset Z Y of size |Z| |Y| without a copy of S.