Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

Ramsey's Theorem states that every infinite graph contains either K∞ or $\overline{K∞}$ as an induced subgraph. For this reason, K∞ and $\overline{K∞}$ are often referred to as the unavoidable infinite graphs. Many similar results have characterized the unavoidable members for various classes of infinite graphs; perhaps the most notable of these is König's Infinity Lemma, which states that every infinite, connected, locally finite graph contains a ray as an induced subgraph. From these findings, one can easily deduce that every infinite, connected graph contains an infinite clique, star, or ray as an induced subgraph; the goal of this dissertation is to obtain analogous results for hypergraphs.

This work is split into two main chapters. First, we focus on hypergraphs that contain a vertex of infinite degree, and find that the unavoidable members are obtained from a finite hypergraph by a specific method of replicating and extending edges; this turns out to be a novel class of hypergraphs, which we call recurrent hypergraphs. We subsequently shift our focus to connected, locally finite hypergraphs, and find that every such hypergraph contains a generalization of a ray as an induced subhypergraph; we call such hypergraphs quasirays. We also provide versions of our results that pertain to finite hypergraphs.

Date

3-27-2025

Committee Chair

Ding, Guoli

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