On Non-Empty Cross-Intersecting Families

Let 2[n] and ( $$\matrix{{\left[ n \right]} \cr i } $$ ) be the power set collection of all i-subsets {1, 2, …, n}, respectively. We call t (t ≥ 2) families $${{\cal A}_1},{{\cal A}_2}, \ldots ,{{\cal A}_t} \subseteq {2^{\left[ \right]}}$$ cross-intersecting if Ai ∩ Aj ≠ ∅ for any $${A_i} \in {{\cal A}_i}$$ $${A_j} A}_j}$$ with j. show that, k +l, l r 1, c > 0 $${\cal A} \left( {\matrix{{\left[ \right),{\cal B} \right)$$ , A}$$ B}$$ are $$\left( {\matrix{{n - r} {l \right) \le \left| {\cal \right| 1} then $$\left| + c\left| \max \left\{ {\left( {\matrix{n c\left( \right),\left( {k \right)} \right\}.$$ This implies a result Tokushige second author (Theorem 3.1) also yields 2k, non-empty cross-intersecting, $$\sum\limits_{i = 1}^t {\left| {{{\cal A}_i}} k} 1,\,\,t\left( \right\},} which generalizes corresponding Hilton Milner 2. Moreover, extremal attaining two upper bounds above characterized.