Solved. The site's commentary credits GPT-5.4 Pro, prompted by Chojecki, with showing that no number of colors suffices for every such family. The frontmatter standing is derived from the accepted claim page Chojecki's note, accepted on the site's curator's credit. The answer determines that the smallest cardinal asked for does not exist; under Erdős's own wording, Problem 12 of Erdős 1987 (printed p. 227), which asks whether any bound exists, it is a negative answer. A note by gavinsherry (a GitHub gist of 2026-04-27, linked from thread post 5935 and prepared with AI assistance) restates the accepted construction and adds, for every finite $r$, a family of countably infinite subsets of a countable ground set, any two meeting in $0$ or $\aleph_0$ points, that every $r$-coloring leaves with a monochromatic member, built from a nonprincipal ultrafilter and the Ramsey number $R_r(3)$. The addendum gets no claim page: it is a strengthened variant that adds nothing to the site's question beyond Theorem 1 of the accepted note, whose family for finite $r$ already lives on the countable set $[\omega]^2$; thread post 6006 gave a simpler ultrafilter example, which the note's author accepted in post 6016.