We elucidate the relationship between threshold and expectation-threshold of a down-set. Qualitatively, our main result demonstrates that there exist down-sets with polynomial gaps their thresholds expectation-thresholds; in particular, logarithmic gap predictions Kahn–Kalai Talagrand (recently proved by Park–Pham Frankston–Kahn–Narayanan–Park) about up-sets do not apply to down-sets. Quantitat...

‎for a set of non-negative integers~$l$‎, ‎the $l$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $a_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|a_u cap a_v|in l$‎. ‎the bipartite $l$-intersection number is defined similarly when the conditions are considered only for the ver...

fuzzy rough n-ary subhypergroups are introduced and characterized.