Owings' problem: an infinite $A$ with $A+A$ monochromatic in any 2-colouring of $\mathbb{N}$? (Erdős #1199)

Statement

Background

A conjecture of Owings [Ow74], recorded by Erdős [Er80, p.104]; listed as open on erdosproblems.com/1199 (fetched 2026-07-13, status 'open', tagged 'additive combinatorics | ramsey theory'). The classification matters: this cannot be resolved by a finite computation. The known frontier is sharp on both sides. Hindman [Hi79] proved the analogous statement is FALSE for $3$-colourings: there is a $3$-colouring of $\mathbb{N}$ admitting no infinite $A$ with $A+A$ monochromatic, so two colours is the only open case. In the other direction, if one drops the doubled elements and only requires the pairwise sums $a+a'$ with $a\neq a'$ to be monochromatic, the answer is yes for any finite number of colours by Hindman's finite-sums theorem (cf. Erdős #532, erdosproblems.com/532). The entire difficulty is therefore forcing the values $2a$ for $a\in A$ into the same colour class as the pairwise sums. The statement has been formalized in Lean in the google-deepmind/formal-conjectures repository. If the conjecture is true, compactness gives finite thresholds $n(k)$ = least $n$ such that every $2$-colouring of $\{1,\ldots,n\}$ contains a $k$-element $A$ with $A+A$ monochromatic; whether $n(k)$ exists is open for each moderate $k$, and no such thresholds appear to have been published. The attacker's tools: SAT or exhaustive search over $2$-colourings of $\{1,\ldots,n\}$ to establish or refute the existence of small thresholds $n(k)$ and to expose the structure of extremal colourings, plus Lean formalization of the partial results (e.g. Hindman's $3$-colouring construction).

References

Attempts

Investigations · 2