- Series
- Dissertation Defense
- Time
- Friday, November 18, 2011 - 1:00pm for 2 hours
- Location
- Skiles 005
- Speaker
- Noah Streib – School of Mathematics, Georgia Tech
- Organizer
- William T. Trotter

This dissertation has two principal components: the dimension of
posets with planar cover graphs, and the cartesian product of posets
whose cover graphs have hamiltonian cycles that parse into symmetric
chains. Posets of height two can have arbitrarily large dimension.
In 1981, Kelly provided an infinite sequence of planar posets that
shows that the dimension of planar posets can also be arbitrarily
large. However, the height of the posets in this sequence increases
with the dimension. In 2009, Felsner, Li, and Trotter conjectured
that for each integer h \geq 2, there exists a least positive
integer c_h so that if P is a poset having a planar cover graph
(hence P is a planar poset as well) and the height of P is h,
then the dimension of P is at most c_h. In the first principal
component of this dissertation we prove this conjecture. We also give
the best known lower bound for c_h, noting that this lower bound is
far from the upper bound. In the second principal component, we
consider posets with the Hamiltonian Cycle--Symmetric Chain Partition
(HC-SCP) property. A poset of width w has this property if its cover
graph has a Hamiltonian cycle which parses into w symmetric chains.
This definition is motivated by a proof of Sperner's Theorem that uses
symmetric chains, and was intended as a possible method of attack on
the Middle Two Levels Conjecture. We show that the subset lattices
have the HC-SCP property by showing that the class of posets with the
strong HC-SCP property, a slight strengthening of the HC-SCP property,
is closed under cartesian product with a two-element chain.
Furthermore, we show that the cartesian product of any two posets from
this class has the HC-SCP property.