On union-closed families with prescribed number of $k$-sets

Abstract

Fix positive integers $N,k,n$ with $n\ge k$. We seek the minimum number of members of size at least $n$ in a finite family of finite sets closed under union and containing exactly $N$ distinct sets of size $k$. This problem is a specialization of the Leck--Roberts--Simpson weighted conjecture: assign weight one to sets of size at least $n$ and zero to smaller sets. The predicted minimizer consists of the unions of nonempty subfamilies of the first $N$ $k$-subsets of the natural numbers, ordered by their largest elements and, when these agree, by their increasing lists lexicographically. For an integer $t\ge 1$, call the range \[ \binom{n+t-1}{k}<N\le\binom{n+t}{k} \] the $t$-th strip. We prove the layered conjecture throughout the first strip, and throughout the second strip for $k=3$. For arbitrary $k$, we prove the second strip for families of subsets of an $(n+2)$-element set. For $k,t\ge 3$, we prove the $t$-th strip for families of subsets of an $(n+t)$-element set whenever $n\ge(t+1)(k-1)$. With no restriction on the ground set, we prove it for $k\ge 3$ and $t\ge 2$ whenever $n>\frac{5}{2} k^2t$. For sufficiently large $k$, we obtain a sufficient bound of order $k^2t/\log k$, uniformly in $t\ge2$.