16 research outputs found
Ample simplicial complexes
Motivated by potential applications in network theory, engineering and
computer science, we study -ample simplicial complexes. These complexes can
be viewed as finite approximations to the Rado complex which has a remarkable
property of {\it indestructibility,} in the sense that removing any finite
number of its simplexes leaves a complex isomorphic to itself. We prove that an
-ample simplicial complex is simply connected and -connected for
large. The number of vertexes of an -ample simplicial complex satisfies
. We use the probabilistic method to
establish the existence of -ample simplicial complexes with vertexes for
any . Finally, we introduce the iterated Paley simplicial
complexes, which are explicitly constructed -ample simplicial complexes with
nearly optimal number of vertexes
Combinatorics
This is the report on the Oberwolfach workshop on Combinatorics, held 1ā7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, ļ¬nite or countable structures ā discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
From Large to In nite Random Simplicial Complexes.
PhD ThesesRandom simplicial complexes are a natural higher dimensional generalisation to the
models of random graphs from Erd}os and R enyi of the early 60s. Now any topological
question one may like to ask raises a question in probability - i.e. what is the chance
this topological property occurs? Several models of random simplicial complexes have
been intensely studied since the early 00s. This thesis introduces and studies two general
models of random simplicial complexes that includes many well-studied models as a
special case. We study their connectivity and Betti numbers, prove a satisfying duality
relation between the two models, and use this to get a range of results for free in the case
where all probability parameters involved are uniformly bounded. We also investigate
what happens when we move to in nite dimensional random complexes and obtain a
simplicial generalisation of the Rado graph, that is we show the surprising result that
(under a large range of parameters) every in nite random simplicial complexes is isomorphic
to a given countable complex X with probability one. We show that this X is
in fact homeomorphic to the countably in nite ball. Finally, we look at and construct
nite approximations to this complex X, and study their topological properties
Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)
International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference āAlgebras, graphs and ordered setā (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Mauriceās many scientific interests:ā¢ Lattices and ordered setsā¢ Combinatorics and graph theoryā¢ Set theory and theory of relationsā¢ Universal algebra and multiple valued logicā¢ Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum