Cross-intersecting sub-families of hereditary families

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 i ≠ j , 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 ≠ { ∅ } 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 star S of H (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.