Authors: Nicholas Selgas
Faculty Mentor: Dr. Dana Bartosova
College: College of Liberal Arts and Sciences
Hypertournaments are a structure to which you can arrive by combining ideas from two related structures, the tournament and the hypergraph. When equipped with a linear order on their vertices, the hypertournaments form a class with certain properties we’d like to investigate. The two questions of interest to us are as follows: is this class Fraïssé and is this class Ramsey? The former question pertains to the closure of the class with respect to different sorts of embeddings. More specifically, we would like to show that whenever we have something embedded into a hypertournament, then that thing is actually a hypertournament and that for any two hypertournaments, there is a third into which both can be embedded. If our class satisfies these properties and a couple others, we say it’s Fraïssé. The main problem, though, is regarding whether or not our class is Ramsey: that is, whenever we have A embedded in B we can find a C with B embedded in it so that when we color all the copies of A in C by some finite number of colors, we can find a monochromatic copy of B in C.