Cross-intersecting sub-families of hereditary families

• Authors: Peter Borg
• Venue: Journal of Combinatorial Theory Series A
• Year: 2012

Families A"1,A"2,...,A"k of sets are said to be cross-intersecting if for any i and j in {1,2,...,k} with ij, any set in A"i intersects any set in A"j. For a finite set X, let 2^X denote the power set of X (the family of all subsets of X). A family H is said to be hereditary if all subsets of any set in H are in H; so H is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family H{@A} of 2^X and any k=|X|+1, both the sum and the product of sizes of k cross-intersecting sub-families A"1,A"2,...,A"k (not necessarily distinct or non-empty) of H are maxima if A"1=A"2=...=A"k=S for some largest starSofH (a sub-family of H whose sets have a common element). We prove this for the case when H is compressed with respect to an element x of X, and for this purpose we establish new properties of the usual compression operation. As we will show, for the sum, the condition k=|X|+1 is sharp. However, for the product, we actually conjecture that the configuration A"1=A"2=...=A"k=S is optimal for any hereditary H and any k=2, and we prove this for a special case.