Gap-doubling closed family of infinite subsets ($a_{n+1}>2a_n$)

The family $D=\{\{a_1<a_2<\cdots \}:a_{n+1}>2a_n\text{ for all }n\}$ is ordinarily closed and nowhere dense. A failed gap is witnessed by a finite prefix, and any cylinder can be refined by adjoining consecutive large integers that fail the gap. Nevertheless $D$ intersects every infinite-subset cone in continuum many sets: a binary tree chooses two sufficiently large next points inside its reservoir at each step.

A closed nowhere dense family can meet every cone $[M{]}^{\omega }$ in continuum many sets, as the gap-doubling closed family of infinite subsets does. Well-order the cones and choose two fresh candidates in each, permanently reserving one of each colour. The red family meets every cone and its complement contains every reserved blue candidate, so it is not Ramsey. As a subset of the closed nowhere dense family, it is still nowhere dense and has the Baire property in the ordinary infinite-subset topology. This recursion requires no regularity assumption on the continuum cardinal.