arXiv subjects

Andrés Aranda

Publications and source records attributed to Andrés Aranda.

6 recordsLinked to original sources

arXiv2025

Ramsey expansions of metrically homogeneous graphs

We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin's catalogue. 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 coherent EPPA. Our results are a contribution to Nešetřil's classification programme of Ramsey classes and can be seen as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion 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 or the Ramsey property. The main results have numerous consequences for the automorphism groups of the Fraisse limits of the classes. As corollaries, we prove amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).

math.CO↗

arXiv2023

Type-respecting amalgamation and big Ramsey degrees

We give an infinitary extension of the Nešetřil-Rödl theorem for category of relational structures with special type-respecting embeddings.

math.CO↗

arXiv2020

The poset of morphism-extension classes of countable graphs

Let $\mathrm{XY_{L,T}}$ denote the class of countably infinite $L$-structures that satisfy the axioms $T$ and in which all homomorphisms of type X (these could be homomorphisms, monomorphisms, or isomorphisms) between finite substructures of $M$ are restrictions of an endomorphism of $M$ of type Y (for example, an automorphism or a surjective endomorphism). Lockett and Truss introduced 18 such morphism-extension classes for relational structures. For a given pair $L,T$, however, two or more morphism-extension properties may define the same class of structures. In this paper, we establish all equalities and inequalities between morphism-extension classes of countable (undirected, loopless) graphs.

math.CO↗

arXiv2020

The independence number of HH-homogeneous graphs and a classification of MB-homogeneous graphs

We show that the independence number of a countably infinite HH-homogeneous graph that does not contain the Rado graph as a spanning subgraph is finite and present a classification of MB-homogeneous graphs up to bimorphism-equivalence as a consequence.

math.CO↗

arXiv2019

Completing graphs to metric spaces

We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions.

math.CO↗

arXiv2018

Morphism extension classes of countable $L$-colored graphs

In~\cite{Hartman:2014}, Hartman, Hubi\v cka and Ma\v sulović studied the hierarchy of morphism extension classes for finite $L$-colored graphs, that is, undirected graphs without loops where sets of colors selected from $L$ are assigned to vertices and edges. They proved that when $L$ is a linear order, the classes $MH_L$ and $HH_L$ coincide, and the same is true for vertex-uniform finite $L$-colored graphs when $L$ is a diamond. In this paper, we explore the same question for countably infinite $L$-colored graphs. We prove that $MH_L=HH_L$ if and only if $L$ is a linear order.

math.CO↗