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Small order graph-tree Ramsey numbers

R.J. Faudree, C.C. Rousseau, R.H. Schelp
1988 Discrete Mathematics  
With but a few exceptions, the Ramsey number r(G, T) is determined for all connected graphs G with at most five vertices and all trees T.  ...  Complications arise in finding this Ramsey number when H is of arbitrary order or has large maximal degree, even when H is a tree and G is of small order.  ...  Thus for graphs of this paper, G of small order and H an arbitrary tree, the chromatic number of G and the order of H are parameters which affect r(G, H).  ... 
doi:10.1016/0012-365x(88)90200-2 fatcat:yyftggepqbf4bhwmosdjmgzdtq

Ramsey Theory on Infinite Structures and the Method of Strong Coding Trees [article]

Natasha Dobrinen
2020 arXiv   pre-print
Ramsey theory of copies of the Rado graph.  ...  Using forcing as a tool for finite searches has allowed the development of Ramsey theory on such trees, leading to solutions for finite big Ramsey degrees of Henson graphs as well as infinite dimensional  ...  Suppose that K satisfies the Ramsey property (or has finite small Ramsey degrees). Let K be the Fraïssé limit of K with some linear order of its universe in order-type ω.  ... 
arXiv:1909.05985v2 fatcat:66ixjzmsqrgu7eluzq5rzxz77a

Generalized Ramsey numbers through adiabatic quantum optimization

Mani Ranjbar, William G. Macready, Lane Clark, Frank Gaitan
2016 Quantum Information Processing  
For generalized Ramsey numbers r(G,H), the emergent order is characterized by graphs G and H.  ...  optimization; and (ii) determine the Ramsey numbers r(T_m,T_n) for trees of order m,n = 6,7,8, most of which were previously unknown.  ...  We determined all generalized Ramsey numbers for trees of order 6-8, resulting in 1600 tree Ramsey numbers, of which (to the best of our knowledge) 1479 are new.  ... 
doi:10.1007/s11128-016-1363-3 fatcat:hlwq5wydofctjnrf3rsaow7mzq

Some Ramsey–Turán type problems and related questions

R.H. Schelp
2012 Discrete Mathematics  
For which graphs G does there exist a constant 0 < c < 1 such that when H is a graph of order the Ramsey number r(G) with δ(H) > c|H|, then any 2-edge coloring of H contains a monochromatic copy of G?  ...  Another variant is to 2-edge color a replacement for the graph K r(G) by a balanced multipartite graph of approximately the same order with the same consequence, a monochromatic G.  ...  Ramsey number for any tree on n vertices when l = 2k = 2n/3.  ... 
doi:10.1016/j.disc.2011.09.015 fatcat:gjjizn7b6zaqzjeypu5plj5tom

Some Known Results and an Open Problem of Tree - Wheel Graph Ramsey Numbers

Surahmat Surahmat
2005 Limits Journal of Mathematics and Its Applications  
Ramsey numbers. In this paper we list some known results and an open problem on graph Ramsey numbers. In the special cases, we list to determine graph Ramsey numbers for trees versus wheels.  ...  For more information, see nice survey small Ramsey numbers in [21] . Some known results on tree-wheel For combination of tree and wheel, several results have been obtained for wheels.  ...  For given graphs G and H, the graph Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement  ... 
doi:10.12962/j1829605x.v2i2.1371 fatcat:jdtunrb35jexpmpbqad7appn34

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

Natasha Dobrinen
2021 arXiv   pre-print
Exciting recent developments include new methods involving logic, topological Ramsey spaces, and category theory.  ...  The paper concentrates on big Ramsey degrees, presenting their essential structure where known and outlining areas for further development.  ...  The "big" refers to the fact that we require an isomorphic subcopy of an infinite structure in which the number of colors is as small as possible (in contrast to the concept of small Ramsey degree in finite  ... 
arXiv:2110.00655v1 fatcat:yl4tfcwtrng5vlalcpbukys5xe

Page 8417 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
A general method of ordering trees is derived as well, and as its applications we give optimal estimates of the orderings of two types of trees.” 2000m:05158 05C50 15A48 Xu, Changqing [Xu, Chang Qin] (  ...  Summary: “We study the spectrum of the Laplacian for trees and present an approach to ordering trees by the spectral radius of the Laplacian.  ... 

Bipartite Ramsey numbers involving stars, stripes and trees
English

Michalis Christou, Costas S. Iliopoulos, Mirka Miller
2013 Electronic Journal of Graph Theory and Applications  
The Ramsey number R(m, n) is the smallest integer p such that any blue-red colouring of the edges of the complete graph K p forces the appearance of a blue K m or a red K n .  ...  We investigate the appearance of simpler monochromatic graphs such as stripes, stars and trees under a 2-colouring of the edges of a bipartite graph.  ...  The minimum order of a complete graph that must achieve that is known as a Ramsey number.  ... 
doi:10.5614/ejgta.2013.1.2.2 fatcat:vtuybuzul5hq5jlo4kwx6ldlxe

Generalizing p-goodness to ordered graphs [article]

Jeremy F. Alm, Patrick Bahls, Kayla Coffey, Carolyn Langhoff
2020 arXiv   pre-print
It is known that the connected graphs that are p-good for all p are the trees. In this paper, we ask which ordered trees are order-p-good for all p.  ...  Ordered Ramsey numbers are fairly new.  ...  In particular, the collection of connected graphs that are p-good for all p is precisely the collection of trees. Ordered Ramsey numbers Definition 6.  ... 
arXiv:1412.3071v4 fatcat:stxxlnpnsbce7lur5busmj2uzi

Page 2010 of Mathematical Reviews Vol. , Issue 90D [page]

1990 Mathematical Reviews  
H. (1-MEMP) Small order graph-tree Ramsey numbers. Proceedings of the First Japan Conference on Graph Theory and Applications (Hakone, 1986). Discrete Math. 72 (1988), no. 1-3, 119-127.  ...  H. (1-MEMP) Some complete bipartite graph-tree Ramsey numbers. Graph theory in memory of G. A. Dirac (Sandbjerg, 1985), 79-89, Ann. Discrete Math., 41, North-Holland, Amsterdam-New York, 1989.  ... 

Page 8623 of Mathematical Reviews Vol. , Issue 2001M [page]

2001 Mathematical Reviews  
In this paper a number of interesting upper and lower bound formulae are established for the graph-theoretic Ramsey number R(G}", G3), where the graphs G7" and GJ have orders m and n respectively, in terms  ...  , G2) for all 3</<~™m, where F consists of all edge-minimal graphs of order j and minimum degree i.” 2001m:05170 05C55 Haanpaa, Harri (FIN-HUT-TCS; Espoo) A lower bound for a Ramsey number.  ... 

Extremal Theory and Bipartite Graph-Tree Ramsey Numbers [chapter]

P. Erdös, R.J. Faudree, C.C. Rousseau, R.H. Schelp
1988 Annals of Discrete Mathematics  
Also, applications of this to the determination of Ramsey number for arbitrary graphs and trees will be discussed .  ...  It will be show that if B is a bipartite graph with parts of order k and i (k--1), then there exists a positive constant c, such that for any tree T of order n and for any j (0--j _-(k -1)), the Ramsey  ...  Erdős etal t a sparse graph [3] [4] [5] 8] . It would be of interest to know which sparse graphs could replace the trees of Theorem 1 and Theorem 5 without altering the results .  ... 
doi:10.1016/s0167-5060(08)70775-1 fatcat:mggki2g7evblned5htfbq346j4

On the Ramsey Numbers of Trees with Small Diameter

Patrick Bahls, T. Scott Spencer
2011 Graphs and Combinatorics  
We estimate the Ramsey number r(T) = r(T,T) for various trees T, obtaining a precise value for r(T) for a large number of trees of diameter 3.  ...  Furthermore we prove that all trees of diameter 3 are Ramsey unsaturated as defined by Balister, Lehel, and Schelp in their article "Ramsey unsaturated and saturated graphs."  ...  We note that Ramsey numbers of trees versus trees have not been very well investigated. In [2] Burr and Roberts establish the Ramsey number of any number of star versus one another.  ... 
doi:10.1007/s00373-011-1098-y fatcat:kzdu3f6hu5efpk2sspimjydy64

Page 3207 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
Some results in Ramsey theory can be stated in terms of ordered Ramsey numbers. The authors find the values of various ordered Ramsey numbers, e.g.  ...  Ordered Ramsey numbers. (English summary) Discrete Math. 247 (2002), no. 1-3, 79-92. Let D,, D> D, be simple directed graphs with no directed cy- cles.  ... 

Restricted Size Ramsey Number for Matching versus Tree and Triangle Unicyclic Graphs of Order Six

Elda Safitri, Peter John, Denny Riama Silaban
2022 International Journal of Computing Science and Applied Mathematics  
In this paper we give the restricted size Ramsey number for a matching of two edges versus tree and triangle unicyclic graphs of order six.  ...  The restricted size Ramsey number r * (G, H) is the smallest positive integer r * such that there is a graph F , of order r(G, H) and with the size r * , satisfying F arrows (G, H).  ...  This Ramsey number gives the order of a graph F that satisfies F → (2K 2 , H), where H is any tree graphs or triangle unicyclic graphs of order six.  ... 
doi:10.12962/j24775401.v8i1.10817 fatcat:5iqmn3zhavbvzl72ljifp5wvoa
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