The reverse mathematics of Hindman's theorem for sums of exactly two elements

Barbara F. Csima, Damir D. Dzhafarov, Denis R. Hirschfeldt, Carl G. Jockusch, Jr., Reed Solomon, Linda Brown Westrick

Published 2018-04-25Version 1

Hindman's Theorem (HT) states that for every coloring of $\mathbb N$ with finitely many colors, there is an infinite set $H \subseteq \mathbb N$ such that all nonempty sums of distinct elements of $H$ have the same color. The investigation of restricted versions of HT from the computability-theoretic and reverse-mathematical perspectives has been a productive line of research recently. In particular, HT$^{\leqslant n}_k$ is the restriction of HT to sums of at most $n$ many elements, with at most $k$ colors allowed, and HT$^{=n}_k$ is the restriction of HT to sums of \emph{exactly} $n$ many elements and $k$ colors. Even HT$^{\leqslant 2}_2$ appears to be a strong principle, and may even imply HT itself over RCA$_0$. In contrast, HT$^{=2}_2$ is known to be strictly weaker than HT over RCA$_0$, since HT$^{=2}_2$ follows immediately from Ramsey's Theorem for $2$-colorings of pairs. In fact, it was open for several years whether HT$^{=2}_2$ is computably true. We show that HT$^{=2}_2$ and similar results with addition replaced by subtraction and other operations are not provable in RCA$_0$, or even WKL$_0$. In fact, we show that there is a computable instance of HT$^{=2}_2$ such that all solutions can compute a function that is diagonally noncomputable relative to $\emptyset'$. It follows that there is a computable instance of HT$^{=2}_2$ with no $\Sigma^0_2$ solution, which is the best possible result with respect to the arithmetical hierarchy. Furthermore, a careful analysis of the proof of the result above about solutions DNC relative to $\emptyset'$ shows that HT$^{=2}_2$ implies RRT$^{=2}_2$, the Rainbow Ramsey Theorem for $2$-colorings of pairs, over RCA$_0$. The most interesting aspect of our construction of computable colorings as above is the use of an effective version of the Lov\'asz Local Lemma due to Rumyantsev and Shen.