Frankl's Union-Closed Sets Conjecture

Claim: every finite non-empty union-closed family \mathcal{F} ≠ {∅} contains an element contained in at least |\mathcal{F}|/2 members.

Frankl posed it in 1978 (folklore traces to Erdős seminars). Knill showed an element can appear in merely (|F|−1)/log₂|F| sets; Wójcik, Balla–Bollobás–Eccles (≥ 3|F|/8 for large families) and others ground forward. Justin Gilmer's 2022 information-theoretic bombshell proved a constant fraction (≈ 3/5 of the natural threshold via entropy arguments) — the first genuine constant-fraction progress in 44 years, and it immediately spawned a small industry of improvements. Full resolution still awaits; a 2024 claimed proof circulated but did not survive scrutiny.