Projects
I work in extremal and enumerative combinatorics, with applications to theoretical computer science. My projects include:
- (with Zachary Hammersla and Kevin Milans)
Ramsey Numbers of Matchings in Ordered Graphs (in preparation)
- (with Levi Axelrod, Nathan Bickel, Anastasia Halfpap, Alex Parker, and Cole Swain)
Statistics of Maximal Independent Sets in Grid-Like Graphs, preprint (2025), 24 pages.
An independent set \(I\) in a graph \(G\) is maximal if \(I\) is not properly contained in any other independent set of \(G\). The study of maximal independent sets (MIS's) in various graphs is well-established, often focusing upon enumeration of the set of MIS's. For an arbitrary graph \(G\), it is typically quite difficult to understand the number and structure of MIS's in \(G\); however, when \(G\) has regular structure, the problem may be more tractable. One class of graphs for which enumeration of MIS's is fairly well-understood is the rectangular grid graphs \(G_{m\times n}\).
We say a graph is grid-like if it is locally isomorphic to a square grid, though the global structure of such a graph might resemble a surface such as a torus or Möbius strip. We study the properties of MIS's in various types of grid-like graphs, in particular determining parity of the set of MIS's, average size of MIS's, and number of pairwise non-isomorphic MIS's in various grid-like graphs.
See our preprint on Arxiv.
See our poster presented at
JMM 2025 in Seattle, WA.
Also see slides.