55 research outputs found
Strongly intersecting integer partitions
We call a sum a1+a2+β’ β’ β’+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 β€ a2 β€ β’ β’ β’ β€ ak and n = a1 + a2 + β’ β’ β’ + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + β’ β’ β’ + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+β’ β’ β’+ak and b1+b2+β’ β’ β’+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 β€ k β€ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k β₯ 4 or k = 3 β€ n ΜΈβ {6, 7, 8} or k = 2 β€ n β€ 3.peer-reviewe
A cross-intersection theorem for subsets of a set
Two families and of sets are said to be
cross-intersecting if each member of intersects each member of
. For any two integers and with , let
denote the family of all subsets of of
size at most . We show that if ,
, and and
are cross-intersecting, then and equality
holds if and
. 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
Cross-intersecting non-empty uniform subfamilies of hereditary families
A set -intersects a set if and have at least common
elements. A set of sets is called a family. Two families and
are cross--intersecting if each set in
-intersects each set in . A family is hereditary
if for each set in , all the subsets of are in
. The th level of , denoted by
, is the family of -element sets in . A set
in is a base of if for each set in
, is not a proper subset of . Let denote
the size of a smallest base of . We show that for any integers
, , and with , there exists an integer
such that the following holds for any hereditary family
with . If is a
non-empty subfamily of , is a non-empty
subfamily of , and are
cross--intersecting, and is maximum under
the given conditions, then for some set in with , either and ,
or , , , and . This was conjectured by the author for and generalizes well-known
results for the case where is a power set.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1805.0524
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
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