The maximum sum of the sizes of cross t-intersecting separated families

Abstract

For a set $ X $ and an integer $ r\geq 0 $, let $ {X \choose \leq r} $ denote the family of subsets of $ X $ that have at most $ r $ elements. Two families $ \mathcal{A}\subset {X\choose \leq r} $ and $ \mathcal{B}\subset {X\choose \leq s} $ are cross $ t $-intersecting if $ |A\cap B|\geq t $ for all $ A\in\mathcal{A}, B\in\mathcal{B} $. In this paper, we considered the measures of cross $ t $-intersecting families $ \mathcal{A}\subset {X\choose \leq r} $, $ \mathcal{B}\subset {X\choose \leq s} $, then we used this result to obtain the maximum sum of sizes of cross $ t $-intersecting separated families.

Keywords

• finite set
• separated families
• cross intersecting
• generating set method
• the shifting method