Partitioning the Boolean lattice into a minimal number of chains of relatively uniform size
It is shown that for a given \(c\) and sufficiently large \(n\) there exist functions \(e(n)\sim\sqrt{n}/2\) and \(f(n)\sim c\sqrt{n\log n}\) such that the Boolean lattice can be partitioned into \({n\choose n/2}\) chains whose lengths are between \(e(n)\) and \(f(n)\). This result is a step forward towards the question of Füredi, who asked whether or not the Boolean lattice can be partitioned into \({n\choose n/2}\) chains whose lengths are either \(a\) or \(a+1\) for some \(a\). The result is also an extension of the previous result of the present authors, where the chain lengths satisfy only the low bound \(e(n)\).
scientific article; zbMATH DE number 1891300