178 research outputs found
Set families with a forbidden subposet
We asymptotically determine the size of the largest family F of subsets of
{1,...,n} not containing a given poset P if the Hasse diagram of P is a tree.
This is a qualitative generalization of several known results including
Sperner's theorem.Comment: 10 pages, 1 figure, motivation and details expanded, final versio
Forbidden subposet problems for traces of set families
In this paper we introduce a problem that bridges forbidden subposet and
forbidden subconfiguration problems. The sets form a
copy of a poset , if there exists a bijection such that for any the relation implies
. A family of sets is \textit{-free} if
it does not contain any copy of . The trace of a family on a
set is .
We introduce the following notions: is
-trace -free if for any -subset , the family
is -free and is trace -free if it is
-trace -free for all . As the first instances of these problems
we determine the maximum size of trace -free families, where is the
butterfly poset on four elements with and determine the
asymptotics of the maximum size of -trace -free families for
. We also propose a generalization of the main conjecture of the area of
forbidden subposet problems
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
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