A CERTAIN 2-COLORING OF THE REALS.

There is a function F : [c]<ω → {0,1} such that if A ⊆ [c]<ω is uncountable, then {F(a ∪ b) : a, b ∈ A, a ≠ b} = {0, 1}. A corollary is that there is a function f : ℝ → {0, 1} such that if A ⊆ ℝ is uncountable, 2 ≤ k < ω, then both 0 and 1 occur as the value of f at the sum of k distinct elements of A. This was originally proved by Hindman, Leader, and Strauss under CH, and they asked if it holds in general. [ABSTRACT FROM AUTHOR]