Frankl's Conjecture for Subgroup Lattices

Abstract

We show that the subgroup lattice of any finite group satisfies Frankl's Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common technical result used to prove both may be of some independent interest.

Well-Graded Families and the Union-Closed Sets Conjecture

“…However, the family F contains 959 total sets, but each element of X is contained in exactly 479 sets; thus, no element of the minimal three-set X is abundant. 1 Taking a slightly different approach, we will next look at the asymptotic behavior of s n and w n as n grows. As mentioned in [22], for r = log 2 n or r = log 2 n the family 2 [r] ∪ [n] has a density of (1 + o(1)) log 2 n 2n as n → ∞.…”

“…Thus, assume that |L| 3 for any L ∈ [K] R . We then have |L ∩ X| = |L| 3 (i.e., every set in [K] R contains at least three elements of X) which implies that ρ X ([K] R ) 1 2 .…”

“…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.…”