On the arithmetic mean of the size of cross-union families
The celebrated Erdős-Ko-Rado theorem provides an upper bound on the size of an intersecting family; its generalization by \textit{L. Pyber} [J. Comb. Theory, Ser. A 43, 85--90 (1986; Zbl 0614.05001)] deals with the size of cross-intersecting families.\par The families \(\mathcal{F}_{0},\mathcal{F}_{1},\dots,\mathcal{F}_{s}\) of \(k\)-subsets of \([n]\) are cross-union if \(F_{0}\cup F_{1}\cup \dots\cup F_{s}\neq \lbrack n]\) for all choices of \(F_{i}\in \mathcal{F}_{i}.\) \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)] proved that for \(n\leq k(s+1)\) the geometric mean of \(\left\vert \mathcal{F}_{i}\right\vert \) is at most \(\binom{n-1}{k}\). It is conjectured that the same is true for the arithmetic mean for the range \(ks<n<k(s+1)\), \(s>s_{0}(k)\). As a support, the conjecture is proved in several cases.
scientific article; zbMATH DE number 7377542
scientific article