Frankl’s Conjecture for Large Semimodular and Planar Semimodular Lattices

A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ x. Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let m denote the number of nonzero join-irreducible elements of L. It is well-known that L consists of at most 2m elements. Let us say that L is large if it has more than 5·2m−3 elements. It is shown that every large semimodular lattice satisfies Frankl’s conjecture. The second result states that every finite semimodular planar lattice L satisfies Frankl’s conjecture. If, in addition, L has at least four elements and its largest element is join-reducible then there are at least two choices for the above-mentioned f . Given an m-element finite set A = {a1, . . . , am}, m ≥ 3, a family (or, in other words, a set) F of at least two subsets of A, i.e. F ⊆ P (A), is called a union-closed family (over A) if X ∪ Y ∈ F whenever X, Y ∈ F . It was Peter Frankl in 1979 (cf. Frankl [9]) who formulated the following conjecture, now called as Frankl’s conjecture or the union-closed sets conjecture: if F is as above then there exists an element of A which is contained in at least half of the members of F . In spite of at least three dozen papers, cf. the bibliography given in [8], this conjecture is still open. Now let L be a finite lattice. As usual, the set of its nonzero join-irreducible elements will be denoted by J(L). We say that L satisfies (the lattice theoretic version of) Frankl’s conjecture if |L| = 1 or there is an f ∈ J(L) such that for the principal filter ↑f = {x ∈ L : f ≤ x} we have |↑f | ≤ |L|/2. Stanley [17] and Poonen [14] or Abe and Nakano [3] have shown that (the original) Frankl’s conjecture is true if and only if all finite lattices satisfy (the lattice theoretic) Frankl’s conjecture. (For details one can also see [6].) This fact has initiated a series of lattice theoretical results given by Abe and Nakano [1], [2], [3], [4], Herrmann and Langsdorf [13], and Reinhold [15], and two combinatorial results achieved by means of lattices, cf. [6] and [8]. In particular, lower semimodular lattices satisfy Frankl’s conjecture by [15], and the method of [15] makes it clear that the situation for (upper) semimodular lattices is much harder. In fact, it is (and it remains) unknown if semimodular lattices satisfy Frankl’s conjecture. The goal of the present paper is to present two subclasses of the class of finite semimodular lattices such that every Date: April 9, 2008, and revised September 1, 2008.