한
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 computer search. Suk (2017) improved the 80-year-old upper bound to 2n+o(n)2^{n+o(n)}. The Erdős conjecture ES(n)=2n−2+1ES(n) = 2^{n-2}+1 remains open.

Further reading

💡 Explore together, one line at a time(0 contributions)

Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.

What did you notice?

You do not need a complete proof. A small observation can open the next path.

Markdown + KaTeX supported (`$x^2$` inline, `$$\sum_{k=1}^n k$$` display)
0 / 3000 characters

moderation policy. Sign in after submitting if you want to edit or delete this attempt from another device.

Loading…

Related concepts

Related mathematicians