603 research outputs found
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
A discrete isodiametric result: the Erd\H{o}s-Ko-Rado theorem for multisets
There are many generalizations of the Erd\H{o}s-Ko-Rado theorem. We give new
results (and problems) concerning families of -intersecting -element
multisets of an -set and point out connections to coding theory and
classical geometry. We establish the conjecture that for such
a family can have at most members
- …