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Problem workshop

One great problem, three kinds of immersion

From a story to your own conjecture

Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.

  1. 1 · DiscoverMeet the questionLearn when it appeared and why it still holds people’s attention.
  2. 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
  3. 3 · DevelopYou are hereBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Week 18 · ErdősPartial progress
Primary source

Happy Ending Problem — How Many Points to Force a Convex n-gon?

Intermediate· Posed 1935A052473

Problem

Find the smallest integer ES(n)ES(n) such that any ES(n)ES(n) points in general position in the plane contain nn points forming a convex polygon. Known: ES(3)=3,ES(4)=5,ES(5)=9,ES(6)=17ES(3) = 3, ES(4) = 5, ES(5) = 9, ES(6) = 17.

Why it matters

Esther Klein proved in 1933 that 5 points always contain a convex quadrilateral; Erdős and Szekeres generalized — and Klein-Szekeres later married, giving the problem its romantic name. The original 1935 bound is conjectured to be improvable to 2n−2+12^{n-2} + 1.

Progress so far

Szekeres-Peters (2006) confirmed ES(6)=17ES(6)=17 by a computer-assisted proof. Suk's 2n+o(n)2^{n+o(n)} upper bound was later sharpened to ES(n)≤2n+O(nlog⁡n)ES(n)\le2^{n+O(\sqrt{n\log n})}. The first open exact case is ES(7)ES(7), for which the general conjecture predicts 33. A 2025 SAT study excludes only anchored subfamilies of the 33-point case.

Further reading

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