arXiv subjects
Publications and source records attributed to Vince Guingona.
We examine indivisibility for classes of graphs. We show that the class of hereditarily $α$-sparse graphs is indivisible if and only if $α> 2$. Additionally, we show that the following classes of graphs are indivisible: perfect graphs, cographs, and chordal graphs, and the following classes of graphs are not indivisible: threshold graphs, split graphs, and distance-hereditary graphs.
We study the preservation of certain properties under products of classes of finite structures. In particular, we examine indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interacts with the lexicographic product, full product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products. In particular, we show that, under mild assumptions, the products considered in this paper do not yield new classes of theories.
In this paper, we introduce the notion of K-rank, where K is an strong amalgamation Fraisse class. Roughly speaking, the K-rank of a partial type is the number of "copies" of K that can be "independently coded" inside of the type. We study K-rank for specific examples of K, including linear orders, equivalence relations, and graphs. We discuss the relationship of K-rank to other ranks in model theory, including dp-rank and op-dimension (a notion coined by the first author and C. D. Hill in previous work).