69 research outputs found
Sperner type theorems with excluded subposets
Let F be a family of subsets of an n-element set. Sperner's theorem says that if there is no inclusion among the members of F then the largest family under this condition is the one containing all ⌊ frac(n, 2) ⌋-element subsets. The present paper surveys certain generalizations of this theorem. The maximum size of F is to be found under the condition that a certain configuration is excluded. The configuration here is always described by inclusions. More formally, let P be a poset. The maximum size of a family F which does not contain P as a (not-necessarily induced) subposet is denoted by La (n, P). The paper is based on a lecture of the author at the Jubilee Conference on Discrete Mathematics [Banasthali University, January 11-13, 2009], but it was somewhat updated in December 2010. © 2011 Elsevier B.V. All rights reserved
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 Ramsey number . III. N-shaped poset
Given partially ordered sets (posets) and , we
say that contains a copy of if for some injective function and for any , if and only if
. For any posets and , the poset Ramsey number
is the least positive integer such that no matter how the elements
of an -dimensional Boolean lattice are colored in blue and red, there is
either a copy of with all blue elements or a copy of with all red
elements.
We focus on the poset Ramsey number for a fixed poset and an
-dimensional Boolean lattice , as grows large. It is known that
, for positive constants and .
However, there is no poset known, for which , for
. This paper is devoted to a new method for finding upper bounds
on using a duality between copies of and sets of elements
that cover them, referred to as blockers. We prove several properties of
blockers and their direct relation to the Ramsey numbers. Using these
properties we show that , for a poset
with four elements and , such that , ,
, and the remaining pairs of elements are incomparable.Comment: 19 pages, 6 figure
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