Cross-intersecting sub-families of hereditary families

Families A1,A2, . . . ,Ak of sets are said to be cross-intersecting if for any i and j in {1, 2, . . . , k} with i 6= j, any set in Ai intersects any set in Aj . For a nite set X, let 2X 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 6 = {∅} of 2X and any k ≥ |X| + 1, both the sum and the product of sizes of k cross-intersecting sub-families A1,A2, . . . ,Ak (not necessarily distinct or non-empty) of H are maxima if A1 = A2 = · · · = Ak = S for some largest star S of H (a subfamily 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 con guration A1 = A2 = · · · = Ak = S is optimal for any hereditary H and any k ≥ 2, and we prove this for a special case. 1 Basic de nitions and notation Unless otherwise stated, we shall use small letters such as x to denote elements of a set or non-negative integers or functions, capital letters such as X to denote sets, and calligraphic letters such as F to denote families (i.e. sets whose elements are sets themselves). It is to be assumed that sets and families are nite. We call a set A an r-element set, or simply an r-set, if its size |A| is r (i.e. if it contains exactly r elements). For any integer n ≥ 1, the set {1, . . . , n} of the rst n positive integers is denoted by [n]. For a set X, the power set of X (i.e. {A : A ⊆ X}) is denoted by 2 , and the family of all r-element subsets of X is denoted by ( X r ) . A family H is said to be a hereditary family (also called an ideal or a downset) if all the subsets of any set in H are in H. Clearly a family is hereditary if and only if it is a union of power sets. A base of H is a set in H that is not a subset of any other set in H.