37 research outputs found
An improvement of the general bound on the largest family of subsets avoiding a subposet
Let be the maximum size of a family of subsets of not containing as a (weak) subposet, and let be the length of
a longest chain in . The best known upper bound for in terms of
and is due to Chen and Li, who showed that for any fixed .
In this paper we show that for any fixed , improving the best known upper bound. By choosing appropriately, we
obtain that as a corollary, which we show is best
possible for general . We also give a different proof of this corollary by
using bounds for generalized diamonds. We also show that the Lubell function of
a family of subsets of not containing as an induced subposet is
for every .Comment: Corrected mistakes, improved the writing. Also added a result about
the Lubell function with forbidden induced subposets. The final publication
will be available at Springer via http://dx.doi.org/10.1007/s11083-016-9390-
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
Induced and non-induced forbidden subposet problems
The problem of determining the maximum size that a -free
subposet of the Boolean lattice can have, attracted the attention of many
researchers, but little is known about the induced version of these problems.
In this paper we determine the asymptotic behavior of , the maximum
size that an induced -free subposet of the Boolean lattice can have
for the case when is the complete two-level poset or the complete
multi-level poset when all 's either equal 4 or
are large enough and satisfy an extra condition. We also show lower and upper
bounds for the non-induced problem in the case when is the complete
three-level poset . These bounds determine the asymptotics of
for some values of independently of the values of and
Problems in extremal graphs and poset theory
In this dissertation, we present three different research topics and results regarding such topics. We introduce partially ordered sets (posets) and study two types of problems concerning them-- forbidden subposet problems and induced-poset-saturation problems. We conclude by presenting results obtained from studying vertex-identifying codes in graphs.
In studying forbidden subposet problems, we are interested in estimating the maximum size of a family of subsets of the -set avoiding a given subposet. We provide a lower bound for the size of the largest family avoiding the poset, which makes use of error-correcting codes. We also provide and upper and lower bound results for a -uniform hypergraph that avoids a \emph{triangle}. Ferrara et al. introduced the concept of studying the minimum size of a family of subsets of the -set avoiding an induced poset, called induced-poset-saturation. In particular, the authors provided a lower bound for the size of an induced-antichain poset and we improve on their lower bound result.
Let be a graph with vertex set and edge set . For any nonnegative integer , let denote the ball of radius around vertex . For a finite graph , an -vertex-identifying code in is a subset , with the property that , for all distinct and , for all . We study graphs with large symmetric differences and -jumbled graphs and estimate the minimum size of a vertex-identifying code in each graph
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