Set Systems Containing Many Maximal Chains

The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power set $\mathcal{P}(\{1,2,\dots,n\})$? We will show that for each fixed $\alpha>0$ there is a family of $\alpha 2^n$ sets containing $(\alpha+o(1))n!$ such chains, and that this is asymptotically best possible. For smaller set systems we are unable to answer the question. We conjecture that a `tower of cubes' construction is extremal. We finish by mentioning briefly a connection to an extremal problem on posets and a variant of our question for the grid graph.Comment: 5 page

Generating all subsets of a finite set with disjoint unions

If X is an n-element set, we call a family G of subsets of X a k-generator for X if every subset of X can be expressed as a union of at most k disjoint sets in G. Frein, Leveque and Sebo conjectured that for n > 2k, the smallest k-generators for X are obtained by taking a partition of X into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We prove this conjecture for all sufficiently large n when k = 2, and for n a sufficiently large multiple of k when k > 2.Comment: Final version, with some additional explanations added in the proof

Comparable pairs in families of sets

Given a family $\mathcal{F}$ of subsets of $[n]$, we say two sets $A, B \in \mathcal{F}$ are comparable if $A \subset B$ or $B \subset A$. Sperner's celebrated theorem gives the size of the largest family without any comparable pairs. This result was later generalised by Kleitman, who gave the minimum number of comparable pairs appearing in families of a given size. In this paper we study a complementary problem posed by Erd\H{o}s and Daykin and Frankl in the early '80s. They asked for the maximum number of comparable pairs that can appear in a family of $m$ subsets of $[n]$, a quantity we denote by $c(n,m)$. We first resolve an old conjecture of Alon and Frankl, showing that $c(n,m) = o(m^2)$ when $m = n^{\omega(1)} 2^{n/2}$. We also obtain more accurate bounds for $c(n,m)$ for sparse and dense families, characterise the extremal constructions for certain values of $m$, and sharpen some other known results.Comment: 18 page