The Union-Closed Families Conjecture

Student: Serdyukov Innokentiy

Educational Programme: Applied Mathematics and Information Science (Bachelor)

Final Grade: 9

Year of Graduation: 2019

Call a family of sets union-closed if for any two member-sets of this family it also contains their union. Frankl's conjecture states that if union-closed family has at least one nonempty member-set, then there exists an element contained in at least half of the member-sets. This conjecture has remained an open problem for almost 40 years. Looking for some local configurations in the family is the most efficient method of proving satisfiability of special cases of conjecture by this time, such configurations are called FC-families. In this paper generalization of Frankl's conjecture and FC-families are considered, proof of generalized criterion of Frankl-Completeness is provided. Also proof of some bounds on values connected to FC-families is given.

Full text (added May 15, 2019)

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