Martin Joseph Roche Ryan

Ryan Martin

The focus of my research is extremal graph theory and random combinatorial structures. I have also worked in a variety of other areas, including intersecting hypergraphs, the theory of positional games and Ramsey theory. Combinatorics, and particularly graph theory, has a wide variety of applications. My own research in the edit distance of graphs has applications in biology and computer scienc...

Spring 2006 M 690 I : Extremal Graph Theory Scribe : Ryan Martin

Theorem 1.3 (RegLem) For every ǫ > 0 and positive integer m, there exist two integers M(ǫ,m) and N(ǫ,m) with the property that, for every graph G with n ≥ N(ǫ,m) vertices, there exists a partition of the vertex set into k + 1 classes V = V0 + V1 + · · ·+ Vl such that • m ≤ l ≤ M(ǫ,m) • |V0| < ǫn • |V1| = |V2| = · · · = |Vl| • For distinct i, j 6= 0, all but at most ǫl2 of the pairs (Vi, Vj) are...

Ryan Brady

Ryan Roberts

The walls of the Louvre are covered with artistic masterpieces, widely recognized as some of the most treasured, creative works known to man. The unique stroke of a brush or the layering of paint can distinguish these geniuses from the masses. Despite the ability to label these pieces of art as especially creative, it remains unclear how one can truly qualify the defining characteristics of cre...

309-316 Ryan

Acute childhood ataxia is a common cause of presentation to the pediatric emergency room or child neurologist. The primary concern on initial assessment is to exclude serious causes of this clinical syndrome, including central nervous system infections and mass lesions, while recognizing the essentially benign nature of acute ataxia in most children. Childhood ataxia can be diagnostically appro...