The union-closed sets conjecture almost holds for almost all random bipartite graphs

The Journey of the Union-Closed Sets Conjecture

“…The literature on graphs provides a rich selection of natural graph classes, even bipartite ones, that may now serve as test cases for Frankl's conjecture. So far, the conjecture has been verified for chordal bipartite, subcubic, seriesparallel [11] and, in an approximate version, random bipartite graphs [12]. We present some of these results here.…”

“…We say that almost every random bipartite graph has property P if for every ε > 0 there is an N such that, whenever m + n ≥ N , the probability that a random bipartite graph on m + n vertices has P is at least 1 − ε. Theorem 10. [12] Let p ∈ (0, 1) be a fixed edge-probability. For every δ > 0, almost every random bipartite graph satisfies Frankl's conjecture up to δ.…”

The graph formulation of the union-closed sets conjecture

“…Another advantage of our reformulation is that it allows to test Frankl's conjecture in a probabilistic setting: in [3] it is shown that almost every random bipartite graph in the standard G(m, n; p) model satisfies Conjecture 2 up to any given δ > 0. That is, almost every such graph contains in each bipartition class a vertex for which the number of maximal stable sets containing it is at most 1 2 + δ times the total number of maximal stable sets.…”

Monoidal networks

“…sets in S (if |S| is sufficiently large, this can be improved slightly to 2.4|S| log 2 |S| , see [32]). For numerous other results, the reader is referred to each of the contributons cited above, as well as [5,6,10,14,18,[27][28][29].…”