61 research outputs found
Diamond-free Families
Given a finite poset P, we consider the largest size La(n,P) of a family of
subsets of that contains no subposet P. This problem has
been studied intensively in recent years, and it is conjectured that exists for general posets P,
and, moreover, it is an integer. For let \D_k denote the -diamond
poset . We study the average number of times a random
full chain meets a -free family, called the Lubell function, and use it for
P=\D_k to determine \pi(\D_k) for infinitely many values . A stubborn
open problem is to show that \pi(\D_2)=2; here we make progress by proving
\pi(\D_2)\le 2 3/11 (if it exists).Comment: 16 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
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
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
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