31 research outputs found
Boolean algebras and Lubell functions
Let denote the power set of . A collection
\B\subset 2^{[n]} forms a -dimensional {\em Boolean algebra} if there
exist pairwise disjoint sets , all non-empty
with perhaps the exception of , so that \B={X_0\cup \bigcup_{i\in I}
X_i\colon I\subseteq [d]}. Let be the maximum cardinality of a family
\F\subset 2^X that does not contain a -dimensional Boolean algebra.
Gunderson, R\"odl, and Sidorenko proved that where .
In this paper, we use the Lubell function as a new measurement for large
families instead of cardinality. The Lubell value of a family of sets \F with
\F\subseteq \tsupn is defined by h_n(\F):=\sum_{F\in \F}1/{{n\choose |F|}}.
We prove the following Tur\'an type theorem. If \F\subseteq 2^{[n]} contains
no -dimensional Boolean algebra, then h_n(\F)\leq 2(n+1)^{1-2^{1-d}} for
sufficiently large . This results implies , where is an absolute constant independent of and . As a
consequence, we improve several Ramsey-type bounds on Boolean algebras. We also
prove a canonical Ramsey theorem for Boolean algebras.Comment: 10 page
An upper bound on the size of diamond-free families of sets
Let be the maximum size of a family of subsets of
not containing as a (weak) subposet. The diamond poset,
denoted , is defined on four elements with the relations
and . has been studied for many posets; one of the
major open problems is determining .
Studying the average number of sets from a family of subsets of on a
maximal chain in the Boolean lattice has been a fruitful method. We
use a partitioning of the maximal chains and introduce an induction method to
show that , improving on the earlier bound of
by Kramer,
Martin and Young.Comment: Accepted by JCTA. Writing is improved based on the suggestions of
referee
Ramsey numbers for partially-ordered sets
We present a refinement of Ramsey numbers by considering graphs with a
partial ordering on their vertices. This is a natural extension of the ordered
Ramsey numbers. We formalize situations in which we can use arbitrary families
of partially-ordered sets to form host graphs for Ramsey problems. We explore
connections to well studied Tur\'an-type problems in partially-ordered sets,
particularly those in the Boolean lattice. We find a strong difference between
Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the
partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl
Rainbow Ramsey problems for the Boolean lattice
We address the following rainbow Ramsey problem: For posets what is the
smallest number such that any coloring of the elements of the Boolean
lattice either admits a monochromatic copy of or a rainbow copy of
. We consider both weak and strong (non-induced and induced) versions of
this problem. We also investigate related problems on (partial) -colorings
of that do not admit rainbow antichains of size
On diamond-free subposets of the Boolean lattice
The Boolean lattice of dimension two, also known as the diamond, consists of
four distinct elements with the following property: . A
diamond-free family in the -dimensional Boolean lattice is a subposet such
that no four elements form a diamond. Note that elements and may or may
not be related.
There is a diamond-free family in the -dimensional Boolean lattice of size
. In this paper, we prove that any
diamond-free family in the -dimensional Boolean lattice has size at most
. Furthermore, we show that the
so-called Lubell function of a diamond-free family in the -dimensional
Boolean lattice is at most , which is asymptotically best possible.Comment: 23 pages, 10 figures Accepted to Journal of Combinatorial Theory,
Series
On diamond-free subposets of the Boolean lattice
The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: A ⊂ B, C ⊂ D. A diamondfree family in the n-dimensional Boolean lattice is a subposet such that no four elements form a diamond. Note that elements B and C may or may not be related. There is a diamond-free family in the n-dimensional Boolean lattice of size (2 − o(1)) n n/2 . In this paper, we prove that any diamond-free family in the ndimensional Boolean lattice has size at most (2.25 + o(1)) n n/2 . Furthermore, we show that the so-called Lubell function of a diamond-free family in the ndimensional Boolean lattice which contains the empty set is at most 2.25 + o(1), which is asympotically best possible