Galvin’s Conjecture on two-color bound for homeomorphic copies of Q in the reals

Establish that for every positive integer K and every coloring c: [R] → {0,1,...,K−1} of unordered pairs of real numbers, there exists a subset Y ⊆ R that is homeomorphic to the rationals Q such that c uses at most two colors on pairs from Y (i.e., |c“[Y]| ≤ 2).

Background

The paper studies infinite Ramsey theory on the reals and related topological spaces. Classical results (e.g., Sierpiński’s coloring) show that there is no direct analogue of Ramsey’s Theorem for R, as one cannot in general obtain monochromatic uncountable subsets under pair colorings. Galvin formulated a conjecture addressing this limitation by seeking a two-color bound on pairs within subsets homeomorphic to the rationals Q.

Historically, Shelah showed the conjecture is consistent assuming large cardinals, and Raghavan–Todorčević later proved it under strong axioms (e.g., a Woodin or strongly compact cardinal). This paper proves the conjecture in ZFC, thereby settling the previously open problem without additional set-theoretic assumptions.