Problem 46

Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$ ?

Source: erdosproblems.com/46

Number theory
Unit fractions
Ramsey theory

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved. Croot's coloring theorem (Annals of Mathematics 157 (2003)) gives an interval $[2,b^r]$ every $r$ -coloring of which has a monochromatic set with reciprocal sum one, and restricting a coloring of the integers to that interval answers the question; Bloom's theorem on sets of positive upper density gives a second route. The site records "PROVED (LEAN)"; the Lean suffix is a catalog label qualified under Existing formalization below, and no local kernel credit is claimed. The claim pages Croot 2003 and Bloom 2021 record the two results, their postings and the acceptance evidence from which the standing above derives.

Cr03: Croot, III, Ernest S., On a coloring conjecture about unit fractions. Ann. of Math. (2) 157 (2003), no. 2, 545--556; arXiv:math/0311421. Library home: Croot 2003.
Bl21: Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021), v2 (2023); J. Eur. Math. Soc. 27 (2025), 4563--4589. Its Theorem 1 restates Croot's theorem and its Theorem 2 implies it. Library home: Bloom 2021.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), printed p. 36. Library home: Erdős and Graham 1980.
Er77c: Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43--72. Library home: Erdos 1977; the two-class form is quoted on the card from printed pp. 58--59.
Er80: Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115; p. 105 as cited by the site. Library home: Erdos 1980; the $k$ -class conjecture is quoted on the card from printed p. 105.
Er92c: Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34--50. Library home: Erdős 1992; the finite $k$ -color form with $n_k$ is quoted on the card from printed p. 46.
Er95: Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas 2 (1995), 165--186. Library home: Erdős 1995; the passage is item 8 of Part I, p. 6.
Er96b: Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333--335. Not held; no library home.
Er97c: Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I, Algorithms Combin. 13, Springer (1997), 47--67; display (4.4) and the paragraph around it, printed pp. 63--64: "an old conjecture of Graham and myself", "We could never prove this even for $k=2$ ", with no prize printed for it. Library home: Erdős 1997; paged at Erdős 1997, Display (4.4).

Problem 46

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