The problem
For every n ≥ 2, any colouring of {2, …, n} into finitely many classes must contain some class with x, y, and x+y all present. Equivalently: the set {2,…,n} cannot be finitely coloured without a monochromatic Schur-type sum.
The Erdős–Graham Conjecture
historicReferences
History & significance
Posed by Erdős and Graham circa 1980 in their 'Old and New Problems' volume. The conjecture resisted attack because standard Ramsey-theory tools give bounds far too weak. Croot (2003) introduced a circle-method approach via generating functions that cracked it — building on ideas from the proof of the Erdős–Szemerédi sum-product phenomenon over finite fields.
The resolution (human proof)
Proven by Ernest Croot III, 2003 (Annals of Mathematics), using a Fourier-analytic argument that decomposes the problem into exponentially many sub-cases controlled by major arcs of the unit circle. The technique has since been applied to other inverse problems in additive combinatorics, making this a rare case where solving one conjecture created new tools rather than consuming existing ones.