On Non-Empty Cross-Intersecting Families

Abstract

Let 2^[n] and ($\matrix{{\left[ n \right]} \cr i \cr }$) be the power set and the collection of all i-subsets of {1, 2, …, n}, respectively. We call t (t ≥ 2) families ${{\cal A}_1},{{\cal A}_2}, \ldots ,{{\cal A}_t} \subseteq {2^{\left[ n \right]}}$ cross-intersecting if A_i ∩ A_j ≠ ∅ for any ${A_i} \in {{\cal A}_i}$ and ${A_j} \in {{\cal A}_j}$ with i ≠ j. We show that, for n ≥ k +l, l ≥ r ≥ 1, c > 0 and ${\cal A} \subseteq \left( {\matrix{{\left[ n \right]} \cr k \cr } } \right),{\cal B} \subseteq \left( {\matrix{{\left[ n \right]} \cr l \cr } } \right)$, if ${\cal A}$ and ${\cal B}$ are cross-intersecting and $\left( {\matrix{{n - r} \cr {l - r} \cr } } \right) \le \left| {\cal B} \right| \le \left( {\matrix{{n - 1} \cr {l - 1} \cr } } \right)$, then

$$\left| {\cal A} \right| + c\left| {\cal B} \right| \le \max \left\{ {\left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - r} \cr k \cr } } \right) + c\left( {\matrix{{n - r} \cr {l - r} \cr } } \right),\left( {\matrix{{n - 1} \cr {k - 1} \cr } } \right) + c\left( {\matrix{{n - 1} \cr {l - 1} \cr } } \right)} \right\}.$$

This implies a result of Tokushige and the second author (Theorem 3.1) and also yields that, for n ≥ 2k, if ${{\cal A}_1},{{\cal A}_2}, \ldots ,{{\cal A}_t} \subseteq \left( {\matrix{{\left[ n \right]} \cr k \cr } } \right)$ are non-empty cross-intersecting, then

$$\sum\limits_{i = 1}^t {\left| {{{\cal A}_i}} \right| \le \max \left\{ {\left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - k} \cr k \cr } } \right) + t - 1,\,\,t\left( {\matrix{{n - 1} \cr {k - 1} \cr } } \right)} \right\},} $$

which generalizes the corresponding result of Hilton and Milner for t = 2. Moreover, the extremal families attaining the two upper bounds above are also characterized.