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Claim. Every finite coloring of the integers has a monochromatic solution of $1=\sum1/n_i$ with $2\le n_1<\cdots<n_k$, as a consequence of the density theorem. The answer to Problem 46 is yes; the result was first proved by Croot, whose claim page is Croot 2003.

Result. Bloom's Theorem 2 (J. Eur. Math. Soc. 27 (2025), no. 11, 4563--4589; arXiv:2112.03726v2, p. 1; paged at theorem_2) states that every set of positive integers with positive upper density contains a finite subset whose reciprocals sum to one. Among the $r$ color classes of the positive integers under a coloring with $r$ colors, one has upper density at least $1/r$, and so does that class with the integer $1$ removed; the theorem applied to the smaller class gives a finite monochromatic set of integers, all at least $2$, with reciprocal sum one. Bloom states on p. 1 that Theorem 2 implies Croot's coloring theorem, which Bloom quotes as Theorem 1, so the paper claims this consequence. The density theorem itself is the subject of Problem 298, and the library holds a complete rewritten proof of it.

Depends on. The density theorem itself, as recorded on Bloom's accepted claim for Problem 298; this page records its consequence for the coloring question.

Acceptance. Refereed: the paper appeared in the Journal of the European Mathematical Society (submitted 1 February 2022, accepted 11 October 2023, first online 11 July 2024). The site's curator is the claimant, so the site's label and commentary count as no independent review on this page; the curator's credit for the problem goes to Croot. The rewritten proof in the library has no independent review.

Formalization. The linked Lean 3 development by Bloom and Bhavik Mehta is the formalization that Appendix B of the paper describes; at the pinned commit its unit_fractions_upper_density states the density theorem. The Lean 4 file that the site's catalog points at for this problem names Croot as its informal author and is linked from Croot's claim page; its route runs through this density theorem. This corpus has not built or audited either development, so neither is formalized evidence here.