Families A1,A2,...,Ak of sets are said
to be \emph{cross-intersecting} if for any i and j in {1,2,...,k}
with i=j, any set in Ai intersects any set in
Aj. For a finite set X, let 2X denote the \emph{power set of
X} (the family of all subsets of X). A family H is said to be
\emph{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 2X and any k≥∣X∣+1, both the sum
and 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 \emph{star S of
H} (a sub-family of H whose sets have a common
element). We prove this for the case when H is \emph{compressed
with respect to an element x of X}, and for this purpose we establish new
properties of the usual \emph{compression operation}. For the product, we
actually conjecture that the configuration A1=A2=...=Ak=S is optimal for any hereditary H and
any k≥2, and we prove this for a special case too.Comment: 13 page