following statement: For every union-closed family, other than the family containing only the empty set, there exists an element that belongs to at least a fraction of 0.01 of the sets in the family. The proof used probabilistic and entropy methods to obtain such a bound. A conjecture within this paper implied a possible improvement to a lower bound fraction of . The union-closed sets conjecture itself corresponds to the fraction . A few days later, three preprints were posted that established the lower bound fraction of . These were shortly followed by two other preprints increasing the lower bound to . History Péter Frankl stated the conjecture in terms of intersection-closed set families in 1979, and so the conjecture is usually credited to him and is sometimes referred to as the Frankl conjecture. The earliest publication of the union-closed version of the conjecture appears to be by . A history of the work on the conjecture up to 2013 was published by . Notes References External links Frankl's union-closed sets conjecture, the Open Problem Garden. Union-Closed Sets Conjecture (1979). In ''Open Problems – Graph Theory and Combinatorics, collected by D. B. West. Families of sets Conjectures Unsolved problems in mathematics Lattice theory