Families of sets with locally bounded width
The author establishes that if \(F\) is a family of subsets of the set \([n]= \{1,\dots, n\}\) such that for each \(x\in \bigcup F\), the poset \(F_ x= \{U\in F\mid x\in U\}\) (ordered by set inclusion) has width at most \(k\) (i.e., a maximal antichain contains at most \(k\) elements) then \(F\) is locally \(k\)-wide and \(| F|\leq (2k)^{k-1} n\). This result of a quite general nature follows from other results where additional restrictions are placed on the members of \(F\). It improves on simpler bounds obtained in the paper and depends on estimates and counting arguments involving families of closed sets also studied by the author, with the closure operations being the Sperner closures in the filters of \(F\), i.e., the intersections of the members of the greatest Sperner antichains containing elements of these filters in the filters themselves. The proof of the main theorem is an interesting application of the techniques used to prove the theorem, techniques which should prove useful in similar settings elsewhere.
scientific article; zbMATH DE number 644732