Families A1β,A2β,...,Akβ of sets are said
to be \emph{cross-intersecting} if for any i and j in {1,2,...,k}
with iξ =j, any set in Aiβ intersects any set in
Ajβ. 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
Hξ ={β } of 2X and any kβ₯β£Xβ£+1, both the sum
and product of sizes of k cross-intersecting sub-families A1β,A2β,...,Akβ (not necessarily distinct or non-empty) of
H are maxima if A1β=A2β=...=Akβ=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 A1β=A2β=...=Akβ=S is optimal for any hereditary H and
any kβ₯2, and we prove this for a special case too.Comment: 13 page
We call (a1β,β¦,anβ) an \emph{r-partial sequence} if exactly r of
its entries are positive integers and the rest are all zero. For c=(c1β,β¦,cnβ) with 1β€c1ββ€β―β€cnβ, let Sc(r)β
be the set of r-partial sequences (a1β,β¦,anβ) with 0β€aiββ€ciβ for each i in {1,β¦,n}, and let Sc(r)β(1) be the set
of members of Sc(r)β which have a1β=1. We say that (a1β,β¦,anβ) \emph{meets} (b1β,β¦,bmβ) if aiβ=biβξ =0 for some i. Two
sets A and B of sequences are said to be \emph{cross-intersecting} if each
sequence in A meets each sequence in B. Let d=(d1β,β¦,dmβ)
with 1β€d1ββ€β―β€dmβ. Let AβSc(r)β and BβSd(s)β such that A and B are cross-intersecting. We
show that β£Aβ£β£Bβ£β€β£Sc(r)β(1)β£β£Sd(s)β(1)β£ if either c1ββ₯3 and d1ββ₯3 or c=d and r=s=n. We also
determine the cases of equality. We obtain this by proving a general
cross-intersection theorem for \emph{weighted} sets. The bound generalises to
one for kβ₯2 cross-intersecting sets.Comment: 20 pages, submitted for publication, presentation improve
Two families A and B of sets are said to be
cross-intersecting if each member of A intersects each member of
B. For any two integers n and k with 0β€kβ€n, let
(β€k[n]β) denote the family of all subsets of {1,β¦,n} of
size at most k. We show that if Aβ(β€r[m]β),
Bβ(β€s[n]β), and A and
B are cross-intersecting, then β£Aβ£β£Bβ£β€i=0βrβ(iβ1mβ1β)j=0βsβ(jβ1nβ1β), and equality
holds if A={Aβ(β€r[m]β):1βA} and
B={Bβ(β€s[n]β):1βB}. Also, we
generalise this to any number of such cross-intersecting families.Comment: 12 pages, submitted. arXiv admin note: text overlap with
arXiv:1212.695