Let X be a finite set and pβ2X, the power set of X,
satisfying three conditions: (a) p is an ideal in 2X, that is,
if Aβp and BβA, then Bβp; (b) For Aβ2X with β£Aβ£β₯2, Aβp if {x,y}βp for any
x,yβA with xξ =y; (c) {x}βp for every xβX. The
pair (X,p) is called a symmetric system if there is a group
Ξ transitively acting on X and preserving the ideal p. A
family {A1β,A2β,β¦,Amβ}β2X is said to be a
cross-p-family of X if {a,b}βp for any aβAiβ and bβAjβ with iξ =j. We prove that if (X,p) is a
symmetric system and {A1β,A2β,β¦,Amβ}β2X is a
cross-p-family of X, then i=1βmββ£Aiββ£β€{β£Xβ£mΞ±(X,p)βifΒ mβ€Ξ±(X,p)β£Xβ£β,ifΒ mβ₯Ξ±(X,p)β£Xβ£β,β where Ξ±(X,p)=max{β£Aβ£:Aβp}. This generalizes Hilton's theorem on
cross-intersecting families of finite sets, and provides analogs for
cross-t-intersecting families of finite sets, finite vector spaces and
permutations, etc.
Moreover, the primitivity of symmetric systems is introduced to characterize
the optimal families.Comment: 15 page
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
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