New Conjectures for Union-Closed Families

Abstract

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead prove that 2a is an upper bound to the number of sets in a union-closed family on a ground set of n elements where each element is in at most a sets for all a, n ∈ N + . Similarly, one could prove that the minimum number of sets containing the most frequent element in a (non-empty) unionclosed family with m sets and n elements is at least m 2 for any m, n ∈ N + . Formulating these problems as integer programs, we observe that the optimal values we computed do not vary with n. We formalize these observations as conjectures, and show that they are not equivalent to the Frankl conjecture while still having wide-reaching implications if proven true. Finally, we prove special cases of the new conjectures and discuss possible approaches to solve them completely.

Cutting planes for families implying Frankl’s conjecture

“…Corollary 3 combined with the integer programming approach to UC families in in Pulaj, Raymond and Theis [18], provides the background of our method. Fix a UC family A such that ∅ ∈ A.…”

“…The connection between Frankl's conjecture and mathematical programming is well-established in Pulaj, Raymond and Theis [18], where the authors derive the equivalence of the problem with an integer program and investigate related conjectures. Furthermore, given an UC family A, Poonen's Theorem yields a constructive proof to determine if A is FC or Non-FC in the form of a fractional polytope with a potentially exponential number of constraints.…”

Well-Graded Families and the Union-Closed Sets Conjecture

“…We say that x is abundant in F if d(x) 1 2 |F|. The union-closed sets conjecture, originally attributed to P. Frankl [20], states that if F ⊆ P(n) is union-closed, then there must be some x ∈ [n] that is contained in at least half of the sets of F; in other words, there is at least one element in [n] that is abundant in F. Some of the more recent examples of work related to the conjecture are given by [1,8,13,17,18], and for a thorough survey of the various results pertaining to the conjecture, as well as an introduction to many of the techniques used in these results, see [4]. In this work we will explore the connection between the union-closed sets conjecture and union-closed families that have the property of being well-graded.…”