36 research outputs found
Set Systems with Restricted Cross-Intersections and the Minimum Rank of Inclusion Matrices
A set system is L-intersecting if any pairwise intersection size lies in L, where L is some set of s nonnegative integers. The celebrated Frankl-Ray-Chaudhuri-Wilson theorems give tight bounds on the size of an L-intersecting set system on a ground set of size n. Such a system contains at most sets if it is uniform and at most sets if it is nonuniform. They also prove modular versions of these results.
We consider the following extension of these problems. Call the set systems {\em L-cross-intersecting} if for every pair of distinct sets A,B with and for some the intersection size lies in . For any k and for n > n 0 (s) we give tight bounds on the maximum of . It is at most if the systems are uniform and at most if they are nonuniform. We also obtain modular versions of these results.
Our proofs use tools from linear algebra together with some combinatorial ideas. A key ingredient is a tight lower bound for the rank of the inclusion matrix of a set system. The s*-inclusion matrix of a set system on [n] is a matrix M with rows indexed by and columns by the subsets of [n] of size at most s, where if and with , we define M AB to be 1 if and 0 otherwise. Our bound generalizes the well-known result that if , then M has full rank . In a combinatorial setting this fact was proved by Frankl and Pach in the study of null t-designs; it can also be viewed as determining the minimum distance of the Reed-Muller codes
Minimum saturated families of sets
We call a family of subsets of -saturated if it
contains no pairwise disjoint sets, and moreover no set can be added to
while preserving this property (here ).
More than 40 years ago, Erd\H{o}s and Kleitman conjectured that an
-saturated family of subsets of has size at least . It is easy to show that every -saturated family has size at
least , but, as was mentioned by Frankl and Tokushige,
even obtaining a slightly better bound of , for some
fixed , seems difficult. In this note, we prove such a result,
showing that every -saturated family of subsets of has size at least
.
This lower bound is a consequence of a multipartite version of the problem,
in which we seek a lower bound on
where are families of subsets of ,
such that there are no pairwise disjoint sets, one from each family
, and furthermore no set can be added to any of the families
while preserving this property. We show that , which is tight e.g.\ by taking
to be empty, and letting the remaining families be the families
of all subsets of .Comment: 8 page
Exact k -Wise Intersection Theorems
A typical problem in extremal combinatorics is the following. Given a large number n and a set L, find the maximum cardinality of a family of subsets of a ground set of n elements such that the intersection of any two subsets has cardinality in L. We investigate the generalization of this problem, where intersections of more than 2 subsets are considered. In particular, we prove that when k−1 is a power of 2, the size of the extremal k-wise oddtown family is (k−1)(n− 2log2(k−1)). Tight bounds are also found in several other basic case
Sperner systems with restricted differences
Let be a family of subsets of and be a subset of
. We say is an -differencing Sperner system if
for any distinct . Let be a prime
and be a power of . Frankl first studied -modular -differencing
Sperner systems and showed an upper bound of the form
. In this paper, we obtain new upper bounds on
-modular -differencing Sperner systems using elementary -adic analysis
and polynomial method, extending and improving existing results substantially.
Moreover, our techniques can be used to derive new upper bounds on subsets of
the hypercube with restricted Hamming distances. One highlight of the paper is
the first analogue of the celebrated Snevily's theorem in the -modular
setting, which results in several new upper bounds on -modular -avoiding
-intersecting systems. In particular, we improve a result of Felszeghy,
Heged\H{u}s, and R\'{o}nyai, and give a partial answer to a question posed by
Babai, Frankl, Kutin, and \v{S}tefankovi\v{c}.Comment: 22 pages, results in table 1 and section 6.1 improve
Bounds on sets with few distances
We derive a new estimate of the size of finite sets of points in metric
spaces with few distances. The following applications are considered:
(1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform
intersecting families of subsets;
(2) we refine the bound of Delsarte-Goethals-Seidel on the maximum size of
spherical sets with few distances;
(3) we prove a new bound on codes with few distances in the Hamming space,
improving an earlier result of Delsarte.
We also find the size of maximal binary codes and maximal constant-weight
codes of small length with 2 and 3 distances.Comment: 11 page