13,607 research outputs found
Cross-intersecting integer sequences
We call an \emph{-partial sequence} if exactly of
its entries are positive integers and the rest are all zero. For with , let
be the set of -partial sequences with for each in , and let be the set
of members of which have . We say that \emph{meets} if for some . Two
sets and of sequences are said to be \emph{cross-intersecting} if each
sequence in meets each sequence in . Let
with . Let and such that and are cross-intersecting. We
show that if either and or and . 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 cross-intersecting sets.Comment: 20 pages, submitted for publication, presentation improve
Cross-intersecting families and primitivity of symmetric systems
Let be a finite set and , the power set of ,
satisfying three conditions: (a) is an ideal in , that is,
if and , then ; (b) For with , if for any
with ; (c) for every . The
pair is called a symmetric system if there is a group
transitively acting on and preserving the ideal . A
family is said to be a
cross--family of if for any and with . We prove that if is a
symmetric system and is a
cross--family of , then where . This generalizes Hilton's theorem on
cross-intersecting families of finite sets, and provides analogs for
cross--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
Strongly intersecting integer partitions
We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe
Cross-intersecting families of vectors
Given a sequence of positive integers , let
denote the family of all sequences of positive integers
such that for all . Two families of sequences (or vectors),
, are said to be -cross-intersecting if no matter how we
select and , there are at least distinct indices
such that . We determine the maximum value of over all
pairs of - cross-intersecting families and characterize the extremal pairs
for , provided that . The case is
quite different. For this case, we have a conjecture, which we can verify under
additional assumptions. Our results generalize and strengthen several previous
results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and
answers a question of Zhang
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