The indivisibility of the homogeneous Kn-free graphs

AbstractWe will prove that for each n ≥ 3 the homogeneous Kn-free graph Hn is indivisible. That means that for every partition of Hn into two classes R and B there is an isomorphic copy of Hn in R or in B. This extends a result of Komjáth and Rödl [Graphs Combin., 2 (1986), 55–60] who have shown that H3 is indivisible

A note on the indivisibility of the Henson graphs

We show that in contrast to the Rado graph, the Henson graphs are not computably indivisible.Comment: 4 pages. This work also appears as part of the author's Ph.D. thesi

On indivisible structures (Model theoretic aspects of the notion of independence and dimension)

An L-structure M is said to be invisible if for any partition M = X ∨ Y, X or Y contains a copy of M as a substructure. In this note we discuss some examples of indivisible structures and their common properties

The oscillation stability problem for the Urysohn sphere: A combinatorial approach

We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for $\ell_2$ in the context of the Urysohn space \Ur. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.Comment: 19 page

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