Michael Albert

In combinatorial geometry it is convenient to reframe problems discretely. A convex geometry is one such abstraction, defined as the pair (X,L), where X is a finite set and L⊂2^X has the properties

1. X,φ∈Li
2. B∈L⇒A∩B∈L
3. ∀A∈L\{X}there exists p∈X\A such that A∪p∈L.

A set C∈Lis said to be a copoint if for some p∈X,C is maximal inX\{p} and furthermore we say C is attached to p. The correspondence C→α(C) :=p is well defined and surjective. We form a graph(V,E), where Vis the set of copoints in(X,L).We say that{C,D}∈E for some copoints C,D if both α(C)∈D and α(D)∈C. The graph(V,E)is called the copoint graph of the convex geometry (X,L).The collection GC of graphs which are realizable (up to isomorphism) as the copoint graph of some convex geometry (X,L) is the subject of this project. We prove that an infinite family of trees is not contained in GC. Furthermore, we characterize the structure of Hasse diagrams that give rise to a tree as its copoint graph, giving a succinct formula for the number (up to relabeling) of such convex geometries on a set X of size n. We discuss the structure of GC with respect to the graph theoretic notions of ‘join’ and ‘complement’. This work was done as part of the 2019 VERUM program at Valparaiso University by Michael Albert (WWU), Evan Franchere (Reed College), Tenzin Zomkyi (City University of New York). Advised by: Jonathan E. Beagley (Valparaiso University