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

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

The Ramsey Theory of Henson graphs

Analogues of Ramsey's Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author's recent result for the triangle-free Henson graph, we prove that for each $k\ge 4$, the $k$-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey's Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker.Comment: 75 pages. Substantial revisions in the presentation. Submitte

Inclusions Among Mixed-Norm Lebesgue Spaces

A mixed LP norm of a function on a product space is the result of successive classical Lp norms in each variable, potentially with a different exponent for each. Conditions to determine when one mixed norm space is contained in another are produced, generalizing the known conditions for embeddings of Lp spaces. The two-variable problem (with four Lp exponents, two for each mixed norm) is studied extensively. The problem\u27s ``unpermuted case simply reduces to a question of Lp embeddings. The other, ``permuted case further divides, depending on the values of the Lp exponents. Often, they fit the ``Minkowski case , when Minkowski\u27s integral inequality provides an easy, complete solution. In the ``non-Minkowski case , the solution is determined by the structure of the measures in the component Lp spaces. When no measure is purely atomic, there can be no mixed-norm embedding in the non-Minkowski case, so for such measures the problem is solved. With at least one purely atomic measure, the non-Minkowski case divides further based on the structure of the measures and the values of the exponents. Various necessary conditions and sufficient conditions are found, solving a number of subcases. Other subcases are shown to be genuinely complicated, with their solutions expressed in terms of an optimization problem known to be computationally difficult. With some difficult cases already present in the two-variable problem, it is impractical to cover every case of the multivariable problem, but results are presented which fully solve some cases. When no measure is purely atomic, the multivariable problem is solved by a reduction to the Minkowski case of certain two-variable subproblems. The multivariable problem with unweighted lp spaces has a similar reduction to easy two-variable subproblems. It is conjectured that this applies more generally; that, regardless of the structures of the involved measures, when every permuted two-variable subproblem fits the Minkowski case, the full multivariable mixed norm inclusion must hold