Dissertation Defense - John Kent, PhD Mathematical Sciences

Candidate: John Kent

Program: PhD Mathematical Sciences

Date: Wednesday, April 29, 2026

Time: 12:00 PM

Place: Johnson Center, Meeting Room G (3^rd floor)

Title: Enumerative Results for Alternating Cycles in Posets

Abstract:  For a partially ordered set P = (X, ≤) there exist hypergraphs where the vertices are

the set of ordered tuples of either all incomparable elements of P or all the critical pairs

of P, and the edges are formed by either all the alternating cycles of P or all the strict

alternating cycles of P. The weak chromatic numbers of these hypergraphs are all equal

to the order dimension of P. Here are established upper bounds on the number of strict

alternating cycles a poset P = (X, ≤) can have in terms of n = |X|, the cardinality of the

groundset of P, and the width w. These bounds also apply to the number of hyperedges of

the associated hypergraph H^s(P), with incomparable pairs as vertices and strict alternating

cycles dual to its hyperedges.