31 research outputs found
The method of double chains for largest families with excluded subposets
For a given finite poset , denotes the largest size of a family
of subsets of not containing as a weak subposet. We
exactly determine for infinitely many posets. These posets are
built from seven base posets using two operations. For arbitrary posets, an
upper bound is given for depending on and the size of the
longest chain in . To prove these theorems we introduce a new method,
counting the intersections of with double chains, rather than
chains.Comment: 8 pages, 5 figures. Submitted to The Electronic Journal of Graph
Theory and Applications (EJGTA
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
Forbidden subposet problems in the grid
For posets and , extremal and saturation problems about weak and
strong -free subposets of have been studied mostly in the case is
the Boolean poset , the poset of all subsets of an -element set ordered
by inclusion. In this paper, we study some instances of the problem with
being the grid, and its connections to the Boolean case and to the forbidden
submatrix problem
Poset-free Families and Lubell-boundedness
Given a finite poset , we consider the largest size \lanp of a family
\F of subsets of that contains no subposet . This
continues the study of the asymptotic growth of \lanp; it has been
conjectured that for all , \pi(P):= \lim_{n\rightarrow\infty} \lanp/\nchn
exists and equals a certain integer, . While this is known to be true for
paths, and several more general families of posets, for the simple diamond
poset \D_2, the existence of frustratingly remains open. Here we
develop theory to show that exists and equals the conjectured value
for many new posets . We introduce a hierarchy of properties for
posets, each of which implies , and some implying more precise
information about \lanp. The properties relate to the Lubell function of a
family \F of subsets, which is the average number of times a random full
chain meets \F. We present an array of examples and constructions that
possess the properties