Big Ramsey degrees using parameter spaces

We show that the universal homogeneous partial order has finite big Ramsey degrees and discuss several corollaries. Our proof uses parameter spaces and the Carlson-Simpson theorem rather than (a strengthening of) the Halpern-L\"auchli theorem and the Milliken tree theorem, which are the primary tools used to give bounds on big Ramsey degrees elsewhere (originating from work of Laver and Milliken). This new technique has many additional applications. To demonstrate this, we show that the homogeneous universal triangle-free graph has finite big Ramsey degrees, thus giving a short proof of a recent result of Dobrinen.Comment: 19 pages, 2 figure

Universal intervals in the homomorphism order of digraphs

In this thesis we solve some open problems related to the homomorphism order of digraphs. We begin by introducing the basic concepts of graphs and homomorphisms and studying some properties of the homomorphism order of digraphs. Then we present the new results. First, we show that the class of digraphs containing cycles has the fractal property (strengthening the density property) . Then we show a density theorem for the class of proper oriented trees. Here we say that a tree is proper if it is not a path. Such result was claimed in 2005 but none proof have been published ever since. We also show that the class of proper oriented trees, in addition to be dense, has the fractal property. We end by considering the consequences of these results and the remaining open questions in this area.Outgoin

Characterizing universal intervals in the homomorphism order of digraphs

In this thesis we characterize all intervals in the homomorphism order of digraphs in terms of universality. To do this, we first show that every interval of the class of digraphs containing cycles is universal. Then we focus our interest in the class of oriented trees (digraphs with no cycles). We give a density theorem for the class of oriented paths and a density theorem for the class of oriented trees, and we strengthen these results by characterizing all universal intervals in these classes. We conclude by summarising all statements and characterizing the universal intervals in the class of digraphs. This solves an open problem in the area

Combinatorial Properties of Finite Models

We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite presentation). Extending classical work of Rado (for the random graph), we find a finite presentation for each of the following classes: homogeneous undirected graphs, homogeneous tournaments and homogeneous partially ordered sets. We also give a finite presentation of the rational Urysohn metric space and some homogeneous directed graphs. We survey well known structures that are finitely presented. We focus on structures endowed with natural partial orders and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism orders for various combinatorial objects. We give a new combinatorial proof of the existence of embedding-universal objects for homomorphism-defined classes of structures. This relates countable embedding-universal structures to homomorphism dualities (finite homomorphism-universal structures) and Urysohn metric spaces. Our explicit construction also allows us to show several properties of these structures.Comment: PhD thesis, unofficial version (missing apple font

Graph Relations and Constrained Homomorphism Partial Orders

We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives from relations between graphs and is related to multihomomorphisms. This gives a generalization of surjective homomorphisms and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time. The theory of the graph homomorphism order is well developed, and from it we consider analogous notions defined for orders induced by constrained homomorphisms. We identify corresponding cores, prove or disprove universality, characterize gaps and dualities. We give a new and significantly easier proof of the universality of the homomorphism order by showing that even the class of oriented cycles is universal. We provide a systematic approach to simplify the proofs of several earlier results in this area. We explore in greater detail locally injective homomorphisms on connected graphs, characterize gaps and show universality. We also prove that for every $d\geq 3$ the homomorphism order on the class of line graphs of graphs with maximum degree $d$ is universal