The Erdős–Szekeres conjecture on convex polygons
Is 2^(n−2) + 1 points always enough to force a convex n-gon? Known exactly only up to n = 6.
Inferred from the community field; finer classification has not been reviewed.
Problem statement
Let be the least such that any points in general position in the plane contain in convex position. Conjecture: .
Lean 4
No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.
Background
The problem originates in a question posed by Esther Klein in 1932; the resulting paper by Erdős and Szekeres led to Szekeres and Klein's marriage, hence Erdős's name for it, the happy ending problem.
Known partial results
- Known exactly for ; was established by Szekeres and Peters in 2006 with a computer search.
- Lower bound from an explicit construction of Erdős and Szekeres.
- Suk (2017) proved , closing the gap to a subexponential factor after eight decades of small improvements.
- is the first unknown case and is a finite, fully specified search problem.
- [1]A. Suk, On the Erdős–Szekeres convex polygon problem, J. Amer. Math. Soc. 30 (2017), 1047–1053. link
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
Machine attempts
0 recordedAttempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.
No machine attempts recorded.
Sign in to run an attempt. New accounts start with enough credits for ten.
Discussion
0 notesNotes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.
No notes yet.
Sign in to add a note.