34 research outputs found
Invitation to intersection problems for finite sets
Extremal set theory is dealing with families, . F of subsets of an . n-element set. The usual problem is to determine or estimate the maximum possible size of . F, supposing that . F satisfies certain constraints. To limit the scope of this survey most of the constraints considered are of the following type: any . r subsets in . F have at least . t elements in common, all the sizes of pairwise intersections belong to a fixed set, . L of natural numbers, there are no . s pairwise disjoint subsets. Although many of these problems have a long history, their complete solutions remain elusive and pose a challenge to the interested reader.Most of the paper is devoted to sets, however certain extensions to other structures, in particular to vector spaces, integer sequences and permutations are mentioned as well. The last part of the paper gives a short glimpse of one of the very recent developments, the use of semidefinite programming to provide good upper bound
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