Define g(n) = the minimum N such that any N points in general position in the plane contain n in convex position (forming a convex n-gon). Known: g(3)=3, g(4)=5 (Erdős–Szekeres 1935), g(5)=9 (Makai 1970), g(6)=17 (Szekeres–Peters 2006, computer-assisted). Erdős–Szekeres conjectured g(n) = \(2^{n−2}\) + 1. Open: determine g(n) for n ≥ 7.
Erdős and Szekeres published the original paper in 1935, proving g(n) is finite with bound \(2^{2n}\). Their upper bound stood essentially unchanged until Suk (2016) proved g(n) = \(2^{n+o(n)}\), matching the conjectured lower bound up to subexponential factors. Andrew Suk's breakthrough used the cup-cap decomposition and the polynomial method — the same tool that cracked cap sets and distinct distances.
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