26,478 research outputs found
Uniform s-cross-intersecting families
In this paper we study a question related to the classical Erd\H{o}s-Ko-Rado
theorem, which states that any family of -element subsets of the set in which any two sets intersect, has cardinality at most
.
We say that two non-empty families are are {\it -cross-intersecting}, if for any we have . In this paper we determine the
maximum of for all . This generalizes a result
of Hilton and Milner, who determined the maximum of
for nonempty -cross-intersecting families.Comment: This article was previously a portion of arXiv:1603.00938v1, which
has been spli
On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method
We study the function which denotes the number of maximal
-uniform intersecting families . Improving a
bound of Balogh at al. on , we determine the order of magnitude of
by proving that for any fixed , holds. Our proof is based on Tuza's set pair
approach.
The main idea is to bound the size of the largest possible point set of a
cross-intersecting system. We also introduce and investigate some related
functions and parameters.Comment: 11 page
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
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