Year of Award

2023

Document Type

Dissertation

Degree Type

Doctor of Philosophy (PhD)

Degree Name

Mathematics

Department or School/College

Department of Mathematical Sciences

Committee Chair

Cory Palmer

Commitee Members

Mark Kayll, Emily Stone, Kelly McKinnie, Puck Rombach

Keywords

co-degree, extremal graph theory, generalized Turan problems, hypergraph Turan problems, rainbow Turan problems, Turan problems

Abstract

In this work, we present results in three settings which generalize the classical Turán problem of maximizing edges in an n-vertex graph subject to some forbidden subgraph condition. Namely, we work in the areas of generalized Turán problems, rainbow Turán problems, and positive co-degree problems. Our main contribution to generalized Turán theory is a general supersaturation result. We also obtain a stability result in cases where the “target” graph is a clique, and consider some problems synthesizing general Turán problems with rainbow Turán problems. Our main rainbow Turán result establishes the rainbow Turán number of the 5-edge path P5, confirming the best-known lower bound construction as extremal. Finally, we introduce a new extremal function on r-graphs, the minimum positive co-degree Turán number, denoted co+ex(n,F), which provides a natural variation on minimum co-degree questions (see [48]). In addition to studying co+ex(n, F) for some small 3-graphs F, we obtain a number of general results on the behavior of co+ex(n,F), including the existence of the density limit γ+(F) := lim n→∞ co+ex(n,F) n for all finite families F and a positive co-degree supersaturation result.

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