4 research outputs found
An extremal problem on crossing vectors
For positive integers and , two vectors and from
are called -crossing if there are two coordinates and
such that and . What is the maximum size of
a family of pairwise -crossing and pairwise non--crossing vectors in
? We state a conjecture that the answer is . We prove
the conjecture for and provide weaker upper bounds for .
Also, for all and , we construct several quite different examples of
families of desired size . This research is motivated by a natural
question concerning the width of the lattice of maximum antichains of a
partially ordered set.Comment: Corrections and improvement
Topological minors of cover graphs and dimension
We show that posets of bounded height whose cover graphs exclude a fixed
graph as a topological minor have bounded dimension. This result was already
proven by Walczak. However, our argument is entirely combinatorial and does not
rely on structural decomposition theorems. Given a poset with large dimension
but bounded height, we directly find a large clique subdivision in its cover
graph. Therefore, our proof is accessible to readers not familiar with
topological graph theory, and it allows us to provide explicit upper bounds on
the dimension. With the introduced tools we show a second result that is
supporting a conjectured generalization of the previous result. We prove that
-free posets whose cover graphs exclude a fixed graph as a topological
minor contain only standard examples of size bounded in terms of .Comment: revised versio
Dimension of posets with planar cover graphs excluding two long incomparable chains
It has been known for more than 40 years that there are posets with planar
cover graphs and arbitrarily large dimension. Recently, Streib and Trotter
proved that such posets must have large height. In fact, all known
constructions of such posets have two large disjoint chains with all points in
one chain incomparable with all points in the other. Gutowski and Krawczyk
conjectured that this feature is necessary. More formally, they conjectured
that for every , there is a constant such that if is a poset
with a planar cover graph and excludes , then
. We settle their conjecture in the affirmative. We also discuss
possibilities of generalizing the result by relaxing the condition that the
cover graph is planar.Comment: New section on connections with graph minors, small correction