Set Representations
Set representations encode graphs and hypergraphs through intersection patterns of families of sets. My work studies odd representations of hypergraphs, where parity conditions make Oddtown-type arguments and linear algebra especially useful.
Current Work
- Odd Representations of Hypergraphs — Manuscript in preparation, joint with Griffin Johnston, Jason O’Neill, and G. Ruiz-Navarro.
Background Articles
- Oddtown and Modtown — Intersection conditions, the linear algebra method, and modular variants that provide background for odd representations.
Planned Background Articles
- Set Representations of Graphs — Intersection representations and the size of the ground set, beginning with the Erdős–Goodman–Pósa paper.
- Set Representations of Hypergraphs — Extending the graph perspective to hypergraphs, including the bounded-degree setting studied by Basu, Johnston, Rödl, and Sales.
- Graham–Pollak — Biclique partitions, algebraic lower bounds, and how the classical theorem motivates odd covering problems.
- Odd Coverings — Parity-based covers of graphs and hypergraphs and their relationship to odd set representations, with Odd Covers of Graphs as background.
- Tensor Rank — Rank-one tensor decompositions and the connection between tensor rank and the odd-covering problems studied in our work.
- Circuit Complexity and the Fusion Method — Cover formulations of computational models and their role in circuit lower bounds, introducing Wigderson’s fusion-method viewpoint.
- Odd Representations — How parity changes the representation problem, why Oddtown arguments enter, and the questions motivating our joint work.