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Symmetries and Ramsey properties of trees

W.L. Fouché
1999 Discrete Mathematics  
In this paper we show the extent to which a finite tree of fixed height is a Ramsey object m the class of trees of the same height can be measured by its symmetry group.  ...  It follows that the complete trees are the only Ramsey objects in ~. For some further results on the Ramsey properties of trees, the reader is referred to [1, 2, 6, 7] .  ...  Moreover, for a 9iven tree T of height h, and for r >~r(T), there is some T1 E ~-h, such that, for any tree T2 E ~-h with [1"2, T1] ~ (~, there is some partition Z : [T2, T] -~ [r] with the property  ... 
doi:10.1016/s0012-365x(99)90081-x fatcat:awyk2u4cdfd23ckhcbvzkk5fxi

Page 917 of Mathematical Reviews Vol. , Issue 80C [page]

1980 Mathematical Reviews  
The derivation of these rules is shown, and the relationship between the spectrum of G and the spectra of G, and G_ is explained in terms of the symmetry properties of the adjacency matrix of G.” 80c:05098  ...  It is also surprising that the connected Ramsey number r(Tm>K,), where T,, is a tree on m vertices, is different for differ- ent types of trees. {For the entire collection see MR 80a:05002a.}  ... 

Page 2542 of Mathematical Reviews Vol. , Issue 90E [page]

1990 Mathematical Reviews  
The symmetry of the spectrum of bipartite graphs and of the patterns of the powers of the adjacency matrix, for example, both extend nicely.  ...  (i1-MEMP) Extremal theory and bipartite graph-tree Ramsey numbers. Proceedings of the First Japan Conference on Graph Theory and Applications (Hakone, 1986).  ... 

Page 3160 of Mathematical Reviews Vol. , Issue 90F [page]

1990 Mathematical Reviews  
The nice properties in (iii) and (iv) can be the property of Baire, the Ramsey property, or Lebesgue measurability, quite in- terchangeably.  ...  The authors show that every A} set of reals is Ramsey if and only if every £} set is Ramsey, if and only if for each s and each r there is a P(D*(r))-generic ultrafilter over L[r}[D*].  ... 

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

1999 Mathematical Reviews  
We next discuss the Ramsey properties of a large variety of structures. We begin with posets and then turn to set systems, whereafter we discuss trees and bipartite graphs, respectively.  ...  From the introduction: “In this paper we discuss the role of symmetry in Ramsey theory. A recent survey of Ramsey theory appears in a paper by J.  ... 

Symmetries between two Ramsey properties [article]

Lorenz Halbeisen
2001 arXiv   pre-print
Further we compare two notions of forcing, the Mathias forcing and a dual form of it, and will give some symmetries between them.  ...  In this article we compare the well-known Ramsey property with a dual form of it, the so called dual-Ramsey property (which was suggested first by Carlson and Simpson).  ...  Acknowledgements: I would like to thank Pierre Matet for all the fruitful discussions concerning the results in this paper, and the referee for valuable comments on an earlier version of this paper.  ... 
arXiv:math/0109101v1 fatcat:k2bk3a5iynbnhevegwsum7papa

Symmetries between two Ramsey properties

Lorenz Halbeisen
1998 Archive for Mathematical Logic  
Further we compare two notions of forcing, the Mathias forcing and a dual form of it, and will give some symmetries between them.  ...  In this article we compare the well-known Ramsey property with a dual form of it, the so called dual-Ramsey property (which was suggested first by Carlson and Simpson).  ...  Acknowledgements: I would like to thank Pierre Matet for all the fruitful discussions concerning the results in this paper, and the referee for valuable comments on an earlier version of this paper.  ... 
doi:10.1007/s001530050096 fatcat:zrxmo2enw5a4fdpynb2wdopwwy

From abstract alpha-Ramsey theory to abstract ultra-Ramsey theory [article]

Timothy Trujillo
2016 arXiv   pre-print
We work within the framework of the Alpha-Theory introduced by Benci and Di Nasso. The Alpha-Theory postulates a few natural properties for an infinite "ideal" number α.  ...  The formulation provides an elementary axiomatics for the methods of abstract ultra-Ramsey theory. The main results are Theorem 10, Theorem 57, Theorem 67 and Theorem 73.  ...  By symmetry, if either st(T ) ⊑ st(S) ∈ T or st(S) ⊑ st(T ) ∈ S then S ∩ T is an α-tree.  ... 
arXiv:1601.03831v1 fatcat:ma6jecv7unaf5m3k7jods4tiim

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.  ...  And similarly for the rainbow Ramsey property. Proof. By symmetry it suffices to assume that A is an induced substructure of B. Let A δ − → A C k and γ : B C → ω be k-delta.  ... 
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  ...  Via compactness this then implies that for all finite graphs B and C and k ∈ω, there exists a graph A so that for every colouring of the copies of C in A such that each colour is used at most k times,  ...  The second author was supported by NSERC of Canada Grant # 690404. The third author was supported by NSERC of Canada Grant # 691325.  ... 
arXiv:1411.6678v1 fatcat:2hast4l7uvfgxfglr2qu7qc7f4

Page 31 of Mathematical Reviews Vol. , Issue 82a [page]

1982 Mathematical Reviews  
The author shows that if G is a tree and Y is its set of endpoints, then S( Y) is a path all of whose points have degree 2 in G.  ...  If the condition |X | > min X is dropped, we get the usual Ramsey relation. It is a remarkable result of J. Paris and L.  ... 

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

Page 3011 of Mathematical Reviews Vol. , Issue 91F [page]

1991 Mathematical Reviews  
Properties of classical Ramsey numbers (in fact the difference of consecutive Ramsey numbers) are used to verify the existence of such families of graphs.  ...  (SAR-UPAM) On the number of spanning trees of X,, and K,, ,. Discrete Math. 84 (1990), no. 2, 205-207.  ... 

Page 54 of Mathematical Reviews Vol. 42, Issue 2 [page]

1971 Mathematical Reviews  
., partitioning of a set of objects, and S’ is monochromatic. This paper concerns Ramsey type theorems relating to edge colourings of graphs.  ...  To prove that P(4)=10, the authors display a 4-regular graph on 10 nodes which has no symmetry, and the 15 4- regular graphs on 9 nodes, which all have some symmetry. R. C.  ... 

Ramsey equivalence for asymmetric pairs of graphs [article]

Simona Boyadzhiyska, Dennis Clemens, Pranshu Gupta, Jonathan Rollin
2022 arXiv   pre-print
The rest of the paper focuses on pairs of the form (T,K_t), where T is a tree and K_t is a complete graph.  ...  We make progress on the asymmetric version of this question and identify several non-trivial families of Ramsey equivalent pairs of connected graphs.  ...  We next study Ramsey equivalence for pairs of the form (T, K t ), where T is a tree and t ≥ 3.  ... 
arXiv:2206.03898v1 fatcat:j4cgsutipnctfe4f6l53nlny7u
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