Limits of single-letter entropy methods for union-closed families
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How far can specified single-letter entropy certificates push the guaranteed element frequency in union-closed families? The barrier depends on product-law admissibility and, in a refinement, an iid singleton class and component hiding.
Why it matters and what counts as progress
Why it matters: A method ceiling distinguishes limits of an entropy technique from the true constant in Frankl’s one-half conjecture.
What counts as progress: A proved ceiling under explicit certificate-class assumptions, or a method escaping them; conditional numerical lower-bound constructions do not establish the true union-closed constant.
How the problem developed
- 16 Nov 2022 · Entropy method [1]
Justin Gilmer Established the first positive universal frequency bound for nontrivial union-closed families, at 0.01, using an information-theoretic argument.
- 8 Sept 2026 · Method ceiling [2]
Andrew Moffat, with Claude agents Bounded what specified single-letter entropy certificates can prove: the ceiling is about 0.38310 for classes containing product laws, with a sharper conditional ceiling for a narrower i.i.d. protocol. This is a limit on the method, not on the union-closed constant.
AI contribution
The ceiling of the single-letter entropy method for the union-closed sets conjecture, and a protocol that reaches it
Moffat’s Theorem 3.1 claims to bound any single-letter entropy certificate whose classes contain product laws by c<=1-h(1/sqrt(2))/sqrt(2)=0.383099... . Theorem 3.4 gives a refined c<=0.382885260... ceiling when the iid protocol has a singleton class and other classes admit component hiding. These are limits of specified certificate methods, not bounds on the true union-closed constant. The paper's c>=0.38284 construction remains conditional on numerical hypotheses and is separate.
Sources
- Justin Gilmer, A constant lower bound for the union-closed sets conjecture
- Andrew Moffat, The ceiling of the single-letter entropy method for the union-closed sets conjecture, and a protocol that reaches it
- Moffat: ceiling of the single-letter entropy method
- Union-closed constant release and verification scope
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