lattice, with equality only if the lattice is a Boolean lattice. As shows, this statement about lattices is equivalent to the Frankl conjecture for union-closed sets: each lattice can be translated into a union-closed set family, and each union-closed set family can be translated into a lattice, such that the truth of the Frankl conjecture for the translated object implies the truth of the conjecture for the original object. This lattice-theoretic version of the conjecture is known to be true for several natural subclasses of lattices but remains open in the general case. Graph-theoretic formulation Another equivalent formulation of the union-closed sets conjecture uses graph theory. In an undirected graph, an independent set is a set of vertices no two of which are adjacent to each other; an independent set is maximal if it is not a subset of a larger independent set. In any graph, the "heavy" vertices that appear in more than half of the maximal independent sets must themselves form an independent set. So, if the graph is non-empty, there always exists at least one non-heavy vertex, a vertex that appears in at most half of the maximal independent sets. The graph formulation of the union-closed sets conjecture states that every finite non-empty graph contains two adjacent non-heavy vertices. It is automatically true when the graph contains an odd cycle, because the independent set of all heavy vertices cannot cover all the edges of the cycle. Therefore, the more interesting case of the conjecture is for bipartite graphs, which have no odd cycles. Another equivalent formulation of the conjecture is that, in every bipartite graph, there exist two vertices, one on each side of the bipartition, such that each of these two vertices belongs to at most half of the graph's maximal independent sets. This conjecture is known to hold for chordal bipartite graphs, bipartite series–parallel graphs, and bipartite graphs of maximum degree three. Partial results The conjecture has been proven for many special cases of union-closed set families. In particular, it is known to be true for families of at most 46 sets. families of sets whose union has at most 11 elements. families of sets in which the smallest set has one or two elements. families of at least subsets of an -element set, for some constant , according to an unpublished preprint. In November 2022, a preprint was posted claiming a proof of the