Erdős Problem #303 — Monochromatic Unit-Fraction Equations

The problem asks whether every finite coloring of the positive integers contains distinct monochromatic $a,b,c$ satisfying $1/a=1/b+1/c$. No proposer or date is identified in the retrieved material.

Known results

• Myers and Parrish computed $f_2(2)=60$, where repeated variables are allowed.
• Brown and Rödl (1991) proved $f_2(k)=O(k^6)$ under that non-distinct formulation.
• A recent paper improves this to $f_2(k)\leq 6k(k+1)(k+2)$.
• Boza, Marín, Revuelta, and Sanz (2019) obtained $f_3(k)=O(k^{43})$; these results still do not impose pairwise distinctness.

Current status (as of June 2025): the pairwise-distinct conjecture remains open; only related results for unit-fraction equations permitting repeated variables were found.