Related papers: The classification of homogeneous finite-dimension…

We provide a classification of the homogeneous 3-dimensional permutation structures, i.e. homogeneous structures in a language of 3 linear orders, partially answering a question of Cameron. We also arrive at a natural description of all…

In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders, and this structural viewpoint is taken up here. The majority of this thesis is…

We determine, up to the equivalence of first-order interdefinability, all structures which are first-order definable in the random partial order. It turns out that these structures fall into precisely five equivalence classes. We achieve…

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…

A structure is called homogeneous if every isomorphism between finite substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Ne\v{s}et\v{r}il introduced a relaxed version of homogeneity:…

In this article we study homotopes of finite-dimensional algebras (not necessarily, associative). In the case of associative algebras we study homotopes by methods of Category theory and give description of so-called well-tempered elements…

We classify the finite connected-homogeneous digraphs, as well as the infinite such digraphs with precisely one end. This completes the classification of all the locally finite connected-homogeneous digraphs.

Transfer systems on finite posets have recently been gaining traction as a key ingredient in equivariant homotopy theory. Additionally, they also naturally occur in the data of a model structure. We give a complete characterization of all…

We classify primitive, rank 1, omega-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely many such structures and that all are built out of finitely…

We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…

This paper reproduces the text of a part of the Author's DPhil thesis. It gives a proof of the classification of non-trivial, finite homogeneous geometries of sufficiently high dimension which does not depend on the classification of the…

Let $V$ be a finite relational vocabulary in which no symbol has arity greater than 2. Let $M$ be countable $V$-structure which is homogeneous, simple and 1-based. The first main result says that if $M$ is, in addition, primitive, then it…

For a finite lattice $\Lambda$, $\Lambda$-ultrametric spaces are a convenient language for describing structures equipped with a family of equivalence relations. When $\Lambda$ is finite and distributive, there exists a generic…

The random permutation is the Fra\"iss\'e limit of the class of finite structures with two linear orders. Answering a problem stated by Peter Cameron in 2002, we use a recent Ramsey-theoretic technique to show that there exist precisely 39…

We study finitely homogeneous dependent rosy structures, adapting results of Cherlin, Harrington, and Lachlan proved for $\omega$-stable $\omega$-categorical structures. In particular, we prove that such structures have finite {\th}-rank…

Structure monoids and groups are algebraic invariants of equational varieties. We show how to construct presentations of these objects from coherent categorifications of equational varieties, generalising several results of Dehornoy. We…

This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…

A partial order is called semilinear iff the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable existentially closed semilinear order, which…

Uniformly finite homology is a coarse invariant for metric spaces; in particular, it is a quasi-isometry invariant for finitely generated groups. In this article, we study uniformly finite homology of finitely generated amenable groups and…

We consider the classification problem for several classes of countable structures which are "vertex-transitive", meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that…