A Certain 2-Coloring of the Reals

Abstract

There is a function $F:[\mathfrak{c}]^{\lt\omega}\to\{0,1\}$ such that if $A\subseteq [\mathfrak{c}]^{\lt\omega}$ is uncountable, then $\{F(a\cup b):a,b\in A, a\neq b\}=\{0,1\}$. A corollary is that there is a function $f:\mathbb{R}\to\{0,1\}$ such that if $A\subseteq\mathbb{R}$ is uncountable, $2\leq k\lt\omega$, 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.