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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.
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Week 16 · ClassicalPartial progress
Primary source

Frankl's Union-Closed Conjecture — Is Some Element in Half the Sets?

Intermediate· Posed 1979

Problem

For any finite family F\mathcal{F} closed under union (and not just {∅}\{\emptyset\}), is there always an element contained in at least half the sets of F\mathcal{F}?

Why it matters

Example: F={∅,{1},{1,2},{1,2,3}}\mathcal{F} = \{\emptyset, \{1\}, \{1,2\}, \{1,2,3\}\} — element 1 is in 3 of 4 sets (75%). The union-closed condition feels like it should force some element to be "dominant", but for 40+ years nobody could even prove any positive constant c>0c > 0.

Progress so far

Justin Gilmer (2022-11-17, then at Google) used entropy methods to prove the first absolute constant c≥0.01c \ge 0.01. Within a week, Sawin, Cambie, Chase, and Yuan pushed this to (3−5)/2≈0.38234(3-\sqrt{5})/2 \approx 0.38234. The conjectured value 0.5 is still open.

Further reading

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

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What did you notice?

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

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