Showcase of Undergraduate Research and Creative Endeavors (SOURCE)

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.