9 research outputs found
Ramsey properties and extending partial automorphisms for classes of finite structures
We show that every free amalgamation class of finite structures with relations and (set-valued) functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Neˇsetˇril-R¨odl Theorem and the second and third authors’ Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results solve several conjectures and provide an easy Ramseyness test for many classes of structures
Supersaturation Problem for the Bowtie
The Tur\'an function denotes the maximal number of edges in an
-free graph on vertices. We consider the function , the
minimal number of copies of in a graph on vertices with
edges. The value of has been extensively studied when is
bipartite or colour-critical. In this paper we investigate the simplest
remaining graph , namely, two triangles sharing a vertex, and establish the
asymptotic value of for .Comment: 23 pages, 1 figur
Dual Ramsey properties for classes of algebras
Almost any reasonable class of finite relational structures has the Ramsey
property or a precompact Ramsey expansion. In contrast to that, the list of
classes of finite algebras with the precompact Ramsey expansion is surprisingly
short. In this paper we show that any nontrivial variety (that is, equationally
defined class of algebras) enjoys various \emph{dual} Ramsey properties. We
develop a completely new set of strategies that rely on the fact that left
adjoints preserve the dual Ramsey property, and then treat classes of algebras
as Eilenberg-Moore categories for a monad. We show that finite algebras in any
nontrivial variety have finite dual small Ramsey degrees, and that every finite
algebra has finite dual big Ramsey degree in the free algebra on countably many
free generators. As usual, these come as consequences of ordered versions of
the statements
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