On the arithmetic mean of the size of cross-union families

Abstract

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$ the standard $n$-element set and ${[n]\choose k}$ the collection of all its $k$-subsets. The families $\mathcal F_0, \ldots, \mathcal F_s \subset {[n]\choose k}$ are said to be cross-union if $F_0 \cup \cdots \cup F_s \neq [n]$ for all choices of $F_i \in \mathcal F_i$. It is known [13] that for $n \leq k(s + 1)$ the geometric mean of $|\mathcal F_i|$ is at most ${n - 1\choose k}$. We conjecture that the same is true for the arithmetic mean for the range $ks < n < k(s + 1)$, $s > s_0(k)$ (Conjecture 8.1) and prove this in several cases. The proof for the case $n = ks + 2$ relies on a novel approach, a combination of shifting and Katona’s cyclic permutation method.