Problem 45

Let $k\geq 2$ . Is there an integer $n_k$ such that, if $D=\{ 1<d<n_k : d\mid n_k\}$ , then for any $k$ -colouring of $D$ there is a monochromatic subset $D'\subseteq D$ such that $\sum_{d\in D'}\frac{1}{d}=1$ ?

Source: erdosproblems.com/45

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 $[2,b^k]$ every $k$ -coloring of which contains a monochromatic set with reciprocal sum one, and $n_k$ can be taken to be the least common multiple of the integers up to $b^k$ . The site records "PROVED (LEAN)"; the Lean suffix is a catalog label whose scope is qualified under Existing formalization below, and no local kernel credit is claimed. The claim page Croot 2003 records the result, its postings and the acceptance evidence (refereed publication and the curator's credit) 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. Context: its Theorem 1 restates Croot's coloring theorem and its Theorem 3 gives another admissible $n_k$ . Library home: Bloom 2021.
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.
Gu04: Guy, Richard K., Unsolved problems in number theory, third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp.; B2 "Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers", printed p. 80, the section the site cites: Erdős defines $n_k$ as the smallest integer such that any partition of its proper divisors into $k$ classes has $n_k$ as a sum of distinct divisors from one class, with $n_1=6$ (from $6=1+2+3$ ) and the existence of $n_2$ unproved. This is a closely related divisor-sum form, not the same question: under $d\mapsto n_k/d$ it also colors the divisor $1$ , which corresponds to the term $1/n_k$ that the site's $D$ excludes, so Guy's $n_k$ need not equal the site's (for $k=1$ , $D(6)=\{2,3\}$ has no subset with reciprocal sum one). Existence for every $k$ agrees between the two forms. Library home: Guy 2004.

Problem 45

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