11 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
-cross -intersecting families via necessary intersection points
Given integers and we call families
-cross
-intersecting if for all , , we have
. We obtain a strong generalisation of
the classic Hilton-Milner theorem on cross intersecting families. In
particular, we determine the maximum of for -cross -intersecting families in the
cases when these are -uniform families or arbitrary subfamilies of
. Only some special cases of these results had been proved
before. We obtain the aforementioned theorems as instances of a more general
result that considers measures of -cross -intersecting families. This
also provides the maximum of for
families of possibly mixed uniformities .Comment: 13 page
Families of vectors without antipodal pairs
Some Erd\H{o}s-Ko-Rado type extremal properties of families of vectors from
are considered
The maximum sum of sizes of non-empty pairwise cross-intersecting families
Two families and are cross-intersecting if for any and . We call
families pairwise
cross-intersecting families if and are
cross-intersecting when . Additionally, if for each , then we say that are non-empty pairwise cross-intersecting.
Let be non-empty pairwise
cross-intersecting families with , ,
and , we determine the maximum value of
and characterize all extremal families. This
answers a question of Shi, Frankl and Qian [Combinatorica 42 (2022)] and
unifies results of Frankl and Tokushige [J. Combin. Theory Ser. A 61 (1992)]
and Shi, Frankl and Qian [Combinatorica 42 (2022)]. The key techniques in
previous works cannot be extended to our situation. A result of Kruskal-Katona
is applied to allow us to consider only families whose elements
are the first elements in lexicographic order. We bound
by a function of the last element
(in the lexicographic order) of , introduce the concepts
`-sequential' and `down-up family', and show that has several types
of local convexities.Comment: 31 page
Discrete Geometry and Convexity in Honour of Imre Bárány
This special volume is contributed by the speakers of the Discrete Geometry and
Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference
is to celebrate the 70th birthday and the scientific achievements of professor
Imre Bárány, a pioneering researcher of discrete and convex geometry, topological
methods, and combinatorics. The extended abstracts presented here are written by
prominent mathematicians whose work has special connections to that of professor
Bárány. Topics that are covered include: discrete and combinatorial geometry,
convex geometry and general convexity, topological and combinatorial methods.
The research papers are presented here in two sections. After this preface and a
short overview of Imre Bárány’s works, the main part consists of 20 short but very
high level surveys and/or original results (at least an extended abstract of them)
by the invited speakers. Then in the second part there are 13 short summaries of
further contributed talks.
We would like to dedicate this volume to Imre, our great teacher, inspiring
colleague, and warm-hearted friend
A size-sensitive inequality for cross-intersecting families
Two families A and B, of k-subsets of an n-set are called cross intersecting if A boolean AND B not equal phi for all A,B epsilon b. Strengthening the classical ErclOs-Ko-Rado theorem, Pyber proved that vertical bar A vertical bar vertical bar B vertical bar 2k. In the present paper we sharpen this inequality. We prove that assuming vertical bar B vertical bar >= ((n - 1 k - 1) - (n -1 k -1)) for some 3 <= i <= k + 1 the stronger inequality vertical bar A vertical bar vertical bar B vertical bar <= ((n - 1 k - 1) + (n-i k -i+1)) x ((n - 1 k - 1) - (n -1 k -1)) holds. These inequalities are best possible. We also present a new short proof of Pyber's inequality and a short computation-free proof of an inequality due to Frankl and Tokushige (1992). (C) 2017 Elsevier Ltd. All rights reserved