Problem 303

Is it true that in any finite colouring of the integers there exists a monochromatic solution to

\frac{1}{a}=\frac{1}{b}+\frac{1}{c}

with distinct $a,b,c$ ?

Source: erdosproblems.com/303

Number theory
Unit fractions

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved, in the site's label, PROVED (LEAN) (page last edited 28 December 2025). Two accepted full claims carry the standing: Brown and Rödl's 1991 theorem, refereed and credited by the site, which proves the stronger positive-integer statement, and Yuan's Seed-Prover Lean proof of December 2025, formalized through this corpus's build of a re-proof of its lemmas. The site relabeled the problem PROVED (LEAN) after Alexeev's comment, and its commentary credits Brown and Rödl. The site's discussion carries no further proof claim.

Problem 303

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Proof claims

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