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Algorithmic randomness and Ramsey properties of countable homogeneous structures [article]

Willem L. Fouché
2013 arXiv   pre-print
We study, in the context of algorithmic randomness, the closed amenable subgroups of the symmetric group S_∞ of a countable set.  ...  In this paper we address this problem by investigating a link between the symmetries associated with Ramsey Fraïssé order classes and algorithmic randomness.  ...  Note that, conversely, the age K of a countable homogeneous structure has properties (1) and (2) .  ... 
arXiv:1308.5506v1 fatcat:xiezazy6l5grjgid35677eitbi

The Random Graph [chapter]

Peter J. Cameron
1997 Algorithms and Combinatorics  
), group theory (simple groups, Cayley graphs), Ramsey theory (Ramsey classes of structures) or topological dynamics (extreme amenability) for some of its properties, and topology (the Urysohn space) for  ...  In brief: every Ramsey class which includes a total order in its structure is the age of a homogeneous structure, and the automorphism groups of such structures are extremely amenable: that is, any continuous  ... 
doi:10.1007/978-3-642-60406-5_32 fatcat:e7fxqhokovblfna2hrhvscsb3a

Algebraic and model theoretic methods in constraint satisfaction [article]

Michael Pinsker
2015 arXiv   pre-print
It describes my current research and some of the most important open questions related to it.  ...  This text is related to the tutorials I gave at the Banff International Research Station and within a "Doc-course" at Charles University Prague in the fall of 2014.  ...  We often think of the Ramsey property as an additional property of Fraïssé classes (and indeed, most homogeneous structures are not Ramsey); we do require the property for our methods.  ... 
arXiv:1507.00931v1 fatcat:gdiqkrt7sbhjfjke4mtqx75uua

Martin-Lof randomness, invariant measures and countable homogeneous structurs [article]

Willem L. Fouche
2012 arXiv   pre-print
We use ideas from topological dynamics (amenability), combinatorics (structural Ramsey theory) and model theory (Fraïssé limits) to study closed amenable subgroups G of the symmetric group S_∞ of a countable  ...  Along these lines we study the random elements of S_∞, which are permutations that transform recursively presented universal structures into such structures which are Martin-Löf random.  ...  Any opinion, findings and conclusions or recommendations expressed in this material are those of the author and therefore the NRF does not accept any liability in regard thereto.  ... 
arXiv:1205.0386v1 fatcat:cvlnun5ppfcbzdzipteldkmvgi

Reducts of Ramsey structures [article]

Manuel Bodirsky, Michael Pinsker
2011 arXiv   pre-print
ordered and homogeneous, and has the Ramsey property.  ...  In model theory, we approach the problem of classifying the reducts of countably infinite ordered homogeneous Ramsey structures in a finite language, and certain decidability questions connected with such  ...  Kechris, Pestov and Todorcevic have characterized the Ramsey property of the age of an ordered homogeneous structure by means of extreme amenability in the following theorem.  ... 
arXiv:1105.6073v1 fatcat:cxh2ajuvsfbsbp4jaakep5d6sq

Combinatorial Properties of Metrically Homogeneous Graphs [article]

Matěj Konečný
2018 arXiv   pre-print
This algorithm also implies the extension property for partial automorphisms (EPPA), another combinatorial property of classes of finite structures.  ...  The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain model-theoretic structures.  ...  Structural Ramsey theory Definition 1.7 (Ramsey property). Let L be a finite relational language and C be a class of finite L-structures.  ... 
arXiv:1805.07425v1 fatcat:jia72p5m3vc6hflolgkkeqylvi

Decidability of definability [article]

Manuel Bodirsky, Michael Pinsker, Todor Tsankov
2012 arXiv   pre-print
Examples of structures \Gamma\ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many  ...  Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.  ...  We remark that it is easy to see that the expansion of any homogeneous structure by finitely many constants is again homogeneous, and that the nontrivial part of the proposition concerns the Ramsey property  ... 
arXiv:1012.2381v4 fatcat:gfgrj2mcdzbbzhsrwtsq4am34m

Schaefer's theorem for graphs

Manuel Bodirsky, Michael Pinsker
2011 Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11  
This is achieved by a universal-algebraic approach, which in turn allows us to use structural Ramsey theory.  ...  infinite random graph.  ...  Let ∆ be a homogeneous structure with a linear order which has the property that its set of finite induced substructures is a Ramsey class, and let g : ∆ k → ∆.  ... 
doi:10.1145/1993636.1993724 dblp:conf/stoc/BodirskyP11 fatcat:z7jlzv2wnbbednypdxbr23nszm

Schaefer's Theorem for Graphs

Manuel Bodirsky, Michael Pinsker
2015 Journal of the ACM  
This is achieved by a universal-algebraic approach, which in turn allows us to use structural Ramsey theory.  ...  infinite random graph.  ...  Let ∆ be a homogeneous structure with a linear order which has the property that its set of finite induced substructures is a Ramsey class, and let g : ∆ k → ∆.  ... 
doi:10.1145/2764899 fatcat:2mlvkuglx5euxmbuphjclv3ede

Decidability of Definability

Manuel Bodirsky, Michael Pinsker, Todor Tsankov
2013 Journal of Symbolic Logic (JSL)  
Examples of structures Γ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more  ...  We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure Δ with a finite relational signature whose age is a Ramsey class and determined  ...  We remark that it is easy to see that the expansion of any homogeneous structure by finitely many constants is again homogeneous, and that the nontrivial part of the proposition concerns the Ramsey property  ... 
doi:10.2178/jsl.7804020 fatcat:la3j7odll5fyrnqripall56s5q

Decidability of Definability

Manuel Bodirsky, Michael Pinsker, Todor Tsankov
2011 2011 IEEE 26th Annual Symposium on Logic in Computer Science  
Examples of structures Γ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C -relations, and many more  ...  We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure ∆ with a finite relational signature whose age is a Ramsey class and determined  ...  We remark that it is easy to see that the expansion of any homogeneous structure by finitely many constants is again homogeneous, and that the nontrivial part of the proposition concerns the Ramsey property  ... 
doi:10.1109/lics.2011.11 dblp:conf/lics/BodirskyPT11 fatcat:xob3ubxjqzd5tfaqdm7qukstze

Structural Ramsey Theory and the Extension Property for Partial Automorphisms [article]

Jan Hubička
2020 arXiv   pre-print
We survey recent developments concerning two properties of classes of finite structures: the Ramsey property and the extension property for partial automorphisms (EPPA).  ...  Acknowledgement The author is grateful to David Evans, Gregory Cherlin and Matěj Konečný for number of remarks and suggestions which improved presentation of this paper.  ...  This Ramsey structure can be thought of as H + with an additional "free" or "generic" linear order of vertices. Example (Countable random graph).  ... 
arXiv:2010.05041v1 fatcat:sracoxd7gzbkxhelalepybixgi

Constraint Satisfaction Problems for Reducts of Homogeneous Graphs

Manuel Bodirsky, Barnaby Martin, Michael Pinsker, András Pongrácz
2019 SIAM journal on computing (Print)  
Another classification of this type has been obtained for CSPs where the constraint language is first-order definable over the random (Rado) graph [14] , making use of structural Ramsey theory.  ...  Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite  ...  The structure (Q, <) and the random graph are among the most prominent examples of homogeneous structures.  ... 
doi:10.1137/16m1082974 fatcat:piol3ztb7ffjde4tkpewjxxfze

Ramsey theory of homogeneous structures: current trends and open problems [article]

Natasha Dobrinen
2021 arXiv   pre-print
Cognate areas, including infinite dimensional Ramsey theory of homogeneous structures and partition theory of uncountable structures, are also discussed.  ...  This article highlights historical achievements in the partition theory of countable homogeneous relational structures, and presents recent work, current trends, and open problems.  ...  She thanks Jan Hubička, Dragan Mašulović, Matěj Konečný, Lionel Nguyen Van Thé, and Andy Zucker for valuable feedback on drafts of this paper.  ... 
arXiv:2110.00655v1 fatcat:yl4tfcwtrng5vlalcpbukys5xe

Structural Ramsey Theory With The Kechris-Pestov-Todorcevic Correspondence In Mind

Lionel Nguyen Van Thé
2013 Zenodo  
My work elaborates on the former and concentrates on two themes: structural Ramsey theory and topological dynamics of Polish transformation groups.  ...  The corresponding material is located at one of the intersection points between combinatorics, topological dynamics and logic via the framework of ultrahomogeneous structures and Fraïssé theory.  ...  More precisely, it deals with an interaction between Ramsey-type properties of finite and countable structures and dynamics of topological groups.  ... 
doi:10.5281/zenodo.1471603 fatcat:jeptngaxanfzpa5lvglyzk3ifq
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