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Symmetry and the Ramsey Degrees of Finite Relational Structures

Willem L. Fouché
1999 Journal of combinatorial theory. Series A  
We show for classes of finite relational structures, including graphs, binary posets, and bipartite graphs, how this measure depends on the symmetries of the structure. 1999 Academic Press In this paper  ...  In this paper, we introduce a measure of the extent to which a finite combinatorial structure is a Ramsey object in the class of objects with a similar structure.  ...  The latter structures include graphs and instances of the relational structures that were studied by Nes etr il and Ro dl in [3] . In [2] , the author determined the Ramsey degrees of posets.  ... 
doi:10.1006/jcta.1998.2910 fatcat:e75ff3gsbzfsfdte7wqfhvj4ha

Rainbow Ramsey simple structures

Natasha Dobrinen, Claude Laflamme, Norbert Sauer
2016 Discrete Mathematics  
A relational structure R is rainbow Ramsey if for every finite induced substructure C of R and every colouring of the copies of C with countably many colours, such that each colour is used at most k times  ...  We show that a class of homogeneous binary relational structures generalizing the Rado graph are rainbow Ramsey.  ...  In fact, any Fraïssé class of finite relational structures with free amalgamation and the Ramsey property is rainbow Ramsey, by a similar argument.  ... 
doi:10.1016/j.disc.2016.04.021 fatcat:hckvwt2jeneedjuxv5nq2rgfdy

Rainbow Ramsey simple structures [article]

Natasha Dobrinen, Claude Laflamme, Norbert Sauer
2014 arXiv   pre-print
A relational structure R is rainbow Ramsey if for every finite induced substructure C of R and every colouring of the copies of C with countably many colours, such that each colour is used at most k times  ...  We show that certain ultrahomogenous binary relational structures, for example the Rado graph, are rainbow Ramsey.  ...  The third author was supported by NSERC of Canada Grant # 691325.  ... 
arXiv:1411.6678v1 fatcat:2hast4l7uvfgxfglr2qu7qc7f4

Methods for Solving Extremal Problems in Practice

Michael Frank, Marc Herbstritt
2016 International Conference on Logic Programming  
of the problem structure.  ...  Further more, we would like to offer ways to exploit the general structure of extremal problems in order to develop constraints and symmetry breaking techniques which will, hopefully, improve existing  ...  Moreover, in both the generate and test, and constrain and generate methods, structural knowledge of the problem as well as symmetry breaking techniques have been employed (e.g, [7, 15, 6] ) to facilitate  ... 
doi:10.4230/oasics.iclp.2016.21 dblp:conf/iclp/Frank16 fatcat:sht6zpbcyzfctltnewnkokqxlq

Page 6628 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews  
We then turn to the discussion of unavoidable symmetries in words over finite alphabets.  ...  Finally we discuss the random generation of highly symmetric structures and the way in which these symmetries are preserved under random partitions.”  ... 

Topological dynamics of automorphism group of countably categorical structures [article]

Aleksander Ivanov
2014 arXiv   pre-print
We consider automorphism groups of some countably categorical structures and their precompact expansions.  ...  We prove that automorphism groups of omega-stable omega-categorical structures have metrizable universal minimal flows. We also study amenability of these groups.  ...  By Proposition 4.4 of [25] A has finite Ramsey degree in K if and only if A has finite embedding Ramsey degree in K.  ... 
arXiv:1412.6657v1 fatcat:dolv2cwy5vfzddojl2d6tzogtq

A Categorical Notion of Precompact Expansion [article]

Keegan Dasilva Barbosa
2020 arXiv   pre-print
We also apply our methodology to calculate the big and little Ramsey degrees of the objects in Age(S(n)) for all n≥ 2.  ...  We generalize the notion of relational precompact expansions of Fraïssé classes via functorial means, inspired by the technique outlined by Laflamme, Nguyen Van Thé and Sauer in their paper Partition properties  ...  Acknowledgements The author would like to thank Lionel Nguyen van Thé, Natasha Dobrinen and Wieslaw Kubis for the insightful comments they gave at the 50 years of Set Theory in Toronto conference.  ... 
arXiv:2002.11751v1 fatcat:y65r7xch7vhg5gnnosobnhbedm

Partition properties of the dense local order and a colored version of Milliken's theorem [article]

C. Laflamme, L. Nguyen Van Thé, N. W. Sauer
2008 arXiv   pre-print
We study the finite dimensional partition properties of the countable homogeneous dense local order.  ...  Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of Milliken's theorem on trees.  ...  For K = Q, the Ramsey degrees and the big Ramsey degrees are always finite and can be computed effectively. More precisely, every X ∈ Q is such that t Q (X) = 1.  ... 
arXiv:0710.2885v3 fatcat:bw2a5gm3dfht3m6c3ttr3t3b4u

Ramsey theorem for designs [article]

Jan Hubička, Jaroslav Nešetřil
2017 arXiv   pre-print
We prove that for any choice of parameters k,t,λ the class of all finite ordered designs with parameters k,t,λ is a Ramsey class.  ...  The language L is usually fixed and understood from the context. If set A is finite we call A finite structure (in most of this paper all structures are finite).  ...  Ramsey classes For structures A, B denote by B A the set of all sub-structures of B, which are isomorphic to A.  ... 
arXiv:1705.02989v1 fatcat:eq5jwofctzfz3e3ac3dgxzxqvy

Page 8214 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
(SA-PRTR-AM; Pretoria) Symmetry and the Ramsey degrees of finite relational structures. (English summary) J. Combin. Theory Ser. A 85 (1999), no. 2, 135-147.  ...  The Ramsey degree t(A) of an object A is defined as the smallest natural number (if it exists) with the following property: For each natural number r and for each object B in C, there is a C in C such  ... 

All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms) [article]

Jan Hubička, Jaroslav Nešetřil
2019 arXiv   pre-print
Among others, we find Ramsey lifts of convexly ordered S-metric spaces and prove the Ramsey theorem for finite models (i.e. structures with both functions and relations) thus providing the ultimate generalisation  ...  of the structural Ramsey theorem.  ...  A large part of work was done while the first author had PIMS Postdoctoral Fellow position at University of Calgary under lead of Claude Laflamme, Norbert Sauer and Robert Woodrow.  ... 
arXiv:1606.07979v3 fatcat:jseo5jbklfc6lilc3cbp7jvdre

Computing the Ramsey Number R(4,3,3) using Abstraction and Symmetry breaking [article]

Michael Codish and Michael Frank and Avraham Itzhakov and Alice Miller
2015 arXiv   pre-print
Along the way it is required to first compute the previously unknown set R(3,3,3;13) consisting of 78,892 Ramsey colorings.  ...  This paper presents a methodology based on abstraction and symmetry breaking that applies to solve hard graph edge-coloring problems.  ...  Acknowledgments We thank Stanislaw Radziszowski for his guidance and comments which helped improve the presentation of this paper.  ... 
arXiv:1510.08266v2 fatcat:diko34glxbhs5emmjcskndtuty

Transitive Sets in Euclidean Ramsey Theory [article]

Imre Leader, Paul A. Russell, Mark Walters
2010 arXiv   pre-print
Calling a finite set transitive if its symmetry group acts transitively---in other words, if all points of the set look the same---our conjecture is that the Ramsey sets are precisely the transitive sets  ...  This notion was introduced by Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere.  ...  Let X be a finite transitive set and let k be a positive integer. Let G be the symmetry group of X.  ... 
arXiv:1012.1350v1 fatcat:6gdsgyrxmrg2rfjy43iib5lcde

Book reports

ErvinY. Rodin
1984 Computers and Mathematics with Applications  
Kennedy, and L. V. Quintas Distance Degree Regular Graphs Bolze and H. Harborth The Ramsey Number r(Kq -x,K5) A.  ...  Subgraphs and the Hamiltonian Theme Entringer Girth of Cubic Graphs with Annular Symmetry Erdos Problems and Results in Graph Theory Erdos and S.  ...  Numerical Solution of Algebraic Equations 7. Special Tools 8. Rayleigh-Ritz and Galerkin Methods 9. Two-Point Boundary-Value Problems II  ... 
doi:10.1016/0898-1221(84)90049-x fatcat:jemlnsahlzdollelv6qog5vase

The weak Ramsey property and extreme amenability [article]

Adam Bartoš, Tristan Bice, Keegan Dasilva Barbosa, Wiesław Kubiś
2021 arXiv   pre-print
We extend the Kechris–Pestov–Todorčević correspondence to weak Fraïssé categories and automorphism groups of generic objects. The new ingredient is the weak Ramsey property.  ...  We demonstrate the theory on several examples including monoid categories, the category of almost linear orders, and categories of strong embeddings of trees.  ...  structures and Ramsey-type properties of their finite substructures.  ... 
arXiv:2110.01694v1 fatcat:udwgufx4fjfthesthru5plf2rm
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