Erdős–Szekeres convex-polygon problem

Open

erdos-szekeres-convex-polygon

Statement

Let

\mathrm{ES}(n)

be the least

N

such that any

N

points in general position contain a convex

n

-gon. Prove the conjectured exact value

\mathrm{ES}(n) = 2^{\,n-2} + 1

Current frontier

erdosproblems.com/107, OPEN. Erdős–Szekeres proved the upper bound (1935) and the lower bound

2^{n-2}+1 \leq \mathrm{ES}(n)

(1960), conjecturing equality; Suk (2017), refined by Holmsen–Mojarrad–Pach–Tardos (2020) to

2^{n+O(\sqrt{n\log n})}

, nearly matches. Exact values are known only through

n = 6

(

\mathrm{ES}(6) = 17

When this counts as solved

VALUE-DETERMINATION. PROOF_COMPLETE for

\mathrm{ES}(n) = 2^{n-2}+1

for all

n

, COUNTEREXAMPLE for a violating

n

. BREAKTHROUGH for a new exact value

\mathrm{ES}(n)

n \geq 7

, or the formula on an infinite subfamily.

Classification

value-determination