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Session Title

STEM and Biomedical Research

Document Type

Oral Presentation

Faculty Mentor

Arran Hamm, Ph.D.

College

College of Arts and Sciences

Department

Department of Mathematics

Description

Ramsey Theory, one of the most well-studied branches of Combinatorics, can be paraphrased as the pursuit of "order amongst chaos." The Fundamental Theorem of Ramsey Theory (for graphs) states that, for any two graphs G and H, any large enough red/blue edge-colored complete graph contains a red G or a blue H. The Ramsey number for G and H, then, is the smallest complete graph with this "unavoidability" property. Recently, the star-critical Ramsey number was introduced, which is a slightly sharper measure on the unavoidability property. Our work focused on the generalized fan, which is formed by taking disjoint copies of a fixed graph H and joining each to a vertex. Recently, researchers have investigated Ramsey and star-critical Ramsey numbers involving this kind of graph which motivated our work. We computed both parameters for a type of generalized fan versus a complete graph, a type of generalized fan versus disjoint triangles, and a type of generalized fan versus a complete graph on four vertices.

Previously Presented/Performed?

UNCG Regional Mathematics and Statistics Conference, Greensboro, North Carolina, November 2019; Sixth Annual Showcase of Undergraduate Research and Creative Endeavors (SOURCE), Winthrop University, April 2020

Grant Support?

Supported by an SC INBRE grant from the National Institute for General Medical Sciences (NIH-NIGMS)

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Apr 24th, 12:00 AM

Ramsey and Star-Critical Ramsey Numbers involving Generalized Fans

Ramsey Theory, one of the most well-studied branches of Combinatorics, can be paraphrased as the pursuit of "order amongst chaos." The Fundamental Theorem of Ramsey Theory (for graphs) states that, for any two graphs G and H, any large enough red/blue edge-colored complete graph contains a red G or a blue H. The Ramsey number for G and H, then, is the smallest complete graph with this "unavoidability" property. Recently, the star-critical Ramsey number was introduced, which is a slightly sharper measure on the unavoidability property. Our work focused on the generalized fan, which is formed by taking disjoint copies of a fixed graph H and joining each to a vertex. Recently, researchers have investigated Ramsey and star-critical Ramsey numbers involving this kind of graph which motivated our work. We computed both parameters for a type of generalized fan versus a complete graph, a type of generalized fan versus disjoint triangles, and a type of generalized fan versus a complete graph on four vertices.