The board / number theory / #484
Erdős Problem #484
PROVED LEANdatabase updated 2026-04-15
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Prove that there exists an absolute constant \(c>0\) such that, whenever \(\{1,\ldots,N\}\) is \(k\)-coloured (and \(N\) is large enough depending on \(k\)) then there are at least \(cN\) many integers in \(\{1,\ldots,N\}\) which are representable as a monochromatic sum (that is, \(a+b\) where \(a,b\in \{1,\ldots,N\}\) are in the same colour class and \(a\neq b\)). A conjecture of Roth. Solved by Erdős, Sárközy, and Sós [ESS89], who in fact prove that there are at least \(\frac{N}{2}-O(N^{1-1/2^{k+1}})\) many even numbers which are of this form.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/484.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/484
Attempts
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