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Symmetry and the Ramsey Degrees of Finite Relational Structures

1999
*
Journal of combinatorial theory. Series A
*

We show for classes

doi:10.1006/jcta.1998.2910
fatcat:e75ff3gsbzfsfdte7wqfhvj4ha
*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. ...##
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Rainbow Ramsey simple structures

2016
*
Discrete Mathematics
*

A

doi:10.1016/j.disc.2016.04.021
fatcat:hckvwt2jeneedjuxv5nq2rgfdy
*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. ...##
###
Rainbow Ramsey simple structures
[article]

2014
*
arXiv
*
pre-print

A

arXiv:1411.6678v1
fatcat:2hast4l7uvfgxfglr2qu7qc7f4
*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. ...##
###
Methods for Solving Extremal Problems in Practice

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 ...

##
###
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]

2014
*
arXiv
*
pre-print

We consider automorphism groups

arXiv:1412.6657v1
fatcat:dolv2cwy5vfzddojl2d6tzogtq
*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. ...##
###
A Categorical Notion of Precompact Expansion
[article]

2020
*
arXiv
*
pre-print

We also apply our methodology to calculate

arXiv:2002.11751v1
fatcat:y65r7xch7vhg5gnnosobnhbedm
*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. ...##
###
Partition properties of the dense local order and a colored version of Milliken's theorem
[article]

2008
*
arXiv
*
pre-print

We study

arXiv:0710.2885v3
fatcat:bw2a5gm3dfht3m6c3ttr3t3b4u
*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. ...##
###
Ramsey theorem for designs
[article]

2017
*
arXiv
*
pre-print

We prove that for any choice

arXiv:1705.02989v1
fatcat:eq5jwofctzfz3e3ac3dgxzxqvy
*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. ...##
###
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]

2019
*
arXiv
*
pre-print

Among others, we find

arXiv:1606.07979v3
fatcat:jseo5jbklfc6lilc3cbp7jvdre
*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. ...##
###
Computing the Ramsey Number R(4,3,3) using Abstraction and Symmetry breaking
[article]

2015
*
arXiv
*
pre-print

Along

arXiv:1510.08266v2
fatcat:diko34glxbhs5emmjcskndtuty
*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. ...##
###
Transitive Sets in Euclidean Ramsey Theory
[article]

2010
*
arXiv
*
pre-print

Calling a

arXiv:1012.1350v1
fatcat:6gdsgyrxmrg2rfjy43iib5lcde
*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. ...##
###
Book reports

1984
*
Computers and Mathematics with Applications
*

Kennedy,

doi:10.1016/0898-1221(84)90049-x
fatcat:jemlnsahlzdollelv6qog5vase
*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 ...##
###
The weak Ramsey property and extreme amenability
[article]

2021
*
arXiv
*
pre-print

We extend

arXiv:2110.01694v1
fatcat:udwgufx4fjfthesthru5plf2rm
*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. ...
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