The Erdős–Szekeres number ES(7)
- Slug:
erdos-szekeres-seven - List: Green P45
- Solver: GPT-5.6 Sol
- Status: residue — the published interval was replayed, but the 33-vertex SAT run was unknown
- Area: Discrete and computational geometry
- Sources: Green P45; arXiv:2512.24061; JCTA 222 (2026), 106195
- Started: 2026-08-23
Statement
Let ES(k) be the least integer N such that every set of N points in the plane in general position, meaning no three are collinear, contains k points in convex position. The first unknown value is ES(7). The Erdős–Szekeres conjecture predicts ES(7) = 33.
The published interval is
The lower bound is witnessed by the classical 32-point construction. The upper bound is the k = 7 specialization of the Mojarrad–Vlachos bound. The 2026 Baek–Balko paper proves the conjectured threshold for decomposable point sets and for split polygons, but does not improve either end of the interval for arbitrary point sets.
What would count as a new bound
Exact coordinates for 33 points in general position with no seven in convex position would prove ES(7) >= 34. Any complete proof of ES(7) <= 112 would improve the upper bound; in particular, a complete independently checked UNSAT certificate at 33 would prove ES(7) = 33.
A new 32-point example only reproduces the known lower bound. A point-set certificate must check every seven-subset in exact arithmetic. An empty convex heptagon, or 7-hole, is a different object and does not certify the required property. A restricted-family exclusion, an unfinished SAT run, or a timeout does not change the published interval.