24 research outputs found
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
A combinatorial proof of the extension property for partial isometries
We present a short and self-contained proof of the extension property for
partial isometries of the class of all finite metric spaces.Comment: 7 pages, 1 figure. Minor revision. Accepted to Commentationes
Mathematicae Universitatis Carolina
Metrické prostory se vzdálenostmi z pologrupy
Strukturální Ramseyova teorie je obor na rozmezí kombinatoriky a teorie modelů s hlubokými souvislostmi s dynamickými systémy. Ramseyovskost většiny známých ramseyovských tříd v konečném binárním symetrickém relačním jazyce se dá dokázat s využitím nějaké varianty tzv. shortest path completion (například Sauerovy S-metrické prostory, Conantovy zobecněné metrické prostory, Braunfel- dovy Λ-ultrametrické prostory či Cherlinovy metricky homogenní grafy). V této práci zkoumáme limity shortest path completion. Nabízíme abstrakci - met- rické prostory se vzdálenostmi z pologrupy - pro všechny zmíněné ramseyovské třídy a studujeme ramseyovské expanze a EPPA (extension property for partial automorphisms) této abstrakce. Na tyto výsledky lze také nahlížet jako na důkaz toho, že samotná otázka, které neúplné struktury mají zúplnění v nějaké amal- gamační třídě, je zajímavá a důležitá. Naše výsledky mají i další aplikace (jako například stationary independence relations). Jako důsledek našich obecných vět znovu dokážeme výsledky Hubičky a Nešetřila o Sauerových S-metrických prostorech, výsledky Hubičky, Nešetřila a autora o Conantových generlizovaných metrických prostorech, Braunfeldovy výsledky o Λ-...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 S-metric spaces, Conant's generalised metric spaces, Braunfeld's Λ-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 Hubička and Nešetřil on Sauer's S-metric spaces, results of Hubička, Nešetřil and the author on Conant's generalised metric spaces, Braunfeld's results on Λ-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 Λ-ultrametric...Katedra aplikované matematikyDepartment of Applied MathematicsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic
Conant's generalised metric spaces are Ramsey
We give Ramsey expansions of classes of generalised metric spaces where distances come from a linearly ordered commutative monoid. This complements results of Conant about the extension property for partial automorphisms and extends an earlier result of the first and the last author giving the Ramsey property of convexly ordered -metric spaces. Unlike Conant's approach, our analysis does not require the monoid to be semiarchimedean
Forbidden cycles in metrically homogeneous graphs
Aranda, Bradley-Williams, Hubi\v{c}ka, Karamanlis, Kompatscher, Kone\v{c}n\'y
and Pawliuk recently proved that for every primitive 3-constrained space
of finite diameter from Cherlin's catalogue of metrically
homogeneous graphs there is a finite family of -edge-labelled cycles such that each -edge-labelled graph is a (not necessarily induced) subgraph of
if and only if it contains no homomorphic images of cycles from
. This analysis is a key to showing that the ages of metrically
homogeneous graphs have Ramsey expansions and the extension property for
partial automorphisms.
In this paper we give an explicit description of the cycles in families
. This has further applications, for example, interpreting the
graphs as semigroup-valued metric spaces or homogenizations of
-categorical -edge-labelled graphs.Comment: 24 pages, 2 table
EPPA for two-graphs and antipodal metric spaces
We prove that the class of finite two-graphs has the extension property for
partial automorphisms (EPPA, or Hrushovski property), thereby answering a
question of Macpherson. In other words, we show that the class of graphs has
the extension property for switching automorphisms. We present a short,
self-contained, purely combinatorial proof which also proves EPPA for the class
of integer valued antipodal metric spaces of diameter 3, answering a question
of Aranda et al.
The class of two-graphs is an important new example which behaves differently
from all the other known classes with EPPA: Two-graphs do not have the
amalgamation property with automorphisms (APA), their Ramsey expansion has to
add a graph, it is not known if they have coherent EPPA and even EPPA itself
cannot be proved using the Herwig--Lascar theorem.Comment: 14 pages, 3 figure
Ramsey expansions of metrically homogeneous graphs
We discuss the Ramsey property, the existence of a stationary independence
relation and the coherent extension property for partial isometries (coherent
EPPA) for all classes of metrically homogeneous graphs from Cherlin's
catalogue, which is conjectured to include all such structures. We show that,
with the exception of tree-like graphs, all metric spaces in the catalogue have
precompact Ramsey expansions (or lifts) with the expansion property. With two
exceptions we can also characterise the existence of a stationary independence
relation and the coherent EPPA.
Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's
classification programme of Ramsey classes and as empirical evidence of the
recent convergence in techniques employed to establish the Ramsey property, the
expansion (or lift or ordering) property, EPPA and the existence of a
stationary independence relation. At the heart of our proof is a canonical way
of completing edge-labelled graphs to metric spaces in Cherlin's classes. The
existence of such a "completion algorithm" then allows us to apply several
strong results in the areas that imply EPPA and respectively the Ramsey
property.
The main results have numerous corollaries on the automorphism groups of the
Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity,
existence of universal minimal flows, ample generics, small index property,
21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor
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