9 research outputs found
Bowtie-free graphs and generic automorphisms
We show that the -categorical existentially closed universal
bowtie-free graph of Cherlin-Shelah-Shi admits generic automorphisms in the
sense of Truss. Moreover, we show that this graph is not finitely
homogenisable.Comment: 14 page
Universal graphs with forbidden subgraphs and algebraic closure
We apply model theoretic methods to the problem of existence of countable
universal graphs with finitely many forbidden connected subgraphs. We show that
to a large extent the question reduces to one of local finiteness of an
associated''algebraic closure'' operator. The main applications are new
examples of universal graphs with forbidden subgraphs and simplified treatments
of some previously known cases
Universal graphs with a forbidden subtree
We show that the problem of the existence of universal graphs with specified
forbidden subgraphs can be systematically reduced to certain critical cases by
a simple pruning technique which simplifies the underlying structure of the
forbidden graphs, viewed as trees of blocks. As an application, we characterize
the trees T for which a universal countable T-free graph exists
Constructing universal graphs for induced-hereditary graph properties
Rado constructed a (simple) denumerable graph R with the positive integers as vertex set
with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in
the m'th position of its binary expansion. It is well known that R is a universal graph in the
set Ic of all countable graphs (since every graph in Ic is isomorphic to an induced subgraph
of R).
In this paper we construct graphs which are universal in or for P for di erent inducedhereditary
properties P of countable graphs. Constructions of universal graphs for the graph
properties containing all graphs with colouring-number at most k+1 and k-degenerate graphs
are obtained by restricting the edges of R. Results on the properties of these graphs are given
and relationships between them are explored. This is followed by a general recursive construction
which proves the existence of a countable universal graph for any induced-hereditary
property of countable general graphs. A general construction of universal graphs for products
of properties of graphs is also presented. The paper is concluded by a comparison between
the two types of constructions of universal graphs.Research of the third author was supported by VEGA Grant No. 2/0194/10.http://link.springer.com/journal/12175hb201
Semigroup-valued metric spaces
The structural Ramsey theory is a field on the boundary of combinatorics and
model theory with deep connections to topological dynamics. Most of the known
Ramsey classes in finite binary symmetric relational language can be shown to
be Ramsey by utilizing a variant of the shortest path completion (e.g. Sauer's
-metric spaces, Conant's generalised metric spaces, Braunfeld's
-ultrametric spaces or Cherlin's metrically homogeneous graphs). In
this thesis we explore the limits of the shortest path completion. We offer a
unifying framework --- semigroup-valued metric spaces --- for all the
aforementioned Ramsey classes and study their Ramsey expansions and EPPA (the
extension property for partial automorphisms). Our results can be seen as
evidence for the importance of studying the completion problem for amalgamation
classes and have some further applications (such as the stationary independence
relation).
As a corollary of our general theorems, we reprove results of Hubi\v{c}ka and
Ne\v{s}et\v{r}il on Sauer's -metric spaces, results of Hub\v{c}ka,
Ne\v{s}et\v{r}il and the author on Conant's generalised metric spaces,
Braunfeld's results on -ultrametric spaces and the results of Aranda
et al. on Cherlin's primitive 3-constrained metrically homogeneous graphs. We
also solve several open problems such as EPPA for -ultrametric spaces,
-metric spaces or Conant's generalised metric spaces.
Our framework seems to be universal enough that we conjecture that every
primitive strong amalgamation class of complete edge-labelled graphs with
finitely many labels is in fact a class of semigroup-valued metric spaces.Comment: Master thesis, defended in June 201
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
Automorphism Groups of Homogeneous Structures
A homogeneous structure is a countable (finite or countably infinite) first order structure such that every isomorphism between finitely generated substructures extends to an automorphism of the whole structure. Examples of homogeneous structures include any countable set, the pentagon graph, the random graph, and the linear ordering of the rationals. Countably infinite homogeneous structures are precisely the Fraisse limits of amalgamation classes of finitely generated structures. Homogeneous structures and their automorphism groups constitute the main theme of the thesis.
The automorphism group of a countably infinite structure becomes a Polish group when endowed with the pointwise convergence topology. Thus, using Baire Category one can formulate the following notions. A Polish group has generic automorphisms if it contains a comeagre conjugacy class. A Polish group has ample generics if it has a comeagre diagonal conjugacy class in every dimension. To study automorphism groups of homogeneous structures as topological groups, we examine combinatorial properties of the corresponding amalgamation classes such as the extension property for partial automorphisms (EPPA), the amalgamation property with automorphisms (APA), and the weak amalgamation property. We also explain how these combinatorial properties yield the aforementioned topological properties in the context of homogeneous structures.
The main results of this thesis are the following. In Chapter 3 we show that any free amalgamation class over a finite relational language has Gaifman clique faithful coherent EPPA. Consequently, the automorphism group of the corresponding free homogeneous structure contains a dense locally finite subgroup, and admits ample generics and the small index property. In Chapter 4 we show that the universal bowtie-free countably infinite graph admits generic automorphisms. In Chapter 5 we prove that Philip Hall's universal locally finite group admits ample generics. In Chapter 6 we show that the universal homogeneous ordered graph does not have locally generic automorphisms. Moreover we prove that the universal homogeneous tournament has ample generics if and only if the class of finite tournaments has EPPA