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Local Ramsey numbers for some graphs

Halina Bielak
2003 Discrete Mathematics  
We study the relation between the 2-local Ramsey number R 2-loc (G) and the Ramsey number R(G), where the complement of G is a disjoint union of some complete graphs or G is a disjoint union of some stars  ...  In this paper we study the relation between the 2-local Ramsey number R 2-loc (G) and the Ramsey number R(G), where the complement of G is a disjoint union of some complete graphs or G is a disjoint union  ...  The k-local Ramsey number R k-loc (G) of a graph G is deÿned as the smallest integer n such that K n contains a monochromatic subgraph G for every local k-colouring of K n .  ... 
doi:10.1016/s0012-365x(02)00800-2 fatcat:dznyczgd2fht5poc5aiufxatfu

Relations between 2-local and 2-mean Ramsey numbers for graphs

Halina Bielak
2007 Discrete Mathematics  
We study the relation between the 2-local Ramsey number R 2-loc (G) and the 2-mean Ramsey number R 2-mean (G) for graphs.  ...  We give the exact value of the 2-mean Ramsey number for some infinite families of graphs.  ...  Acknowledgements I would like to thank the anonymous referees for their comments that improved the presentation of this paper.  ... 
doi:10.1016/j.disc.2005.11.036 fatcat:wuk5qyxadrhyrdyg67lz4hlpqy

Local k-colorings of graphs and hypergraphs

A Gyárfás, J Lehel, J Nešetřil, V Rödl, R.H Schelp, Zs Tuza
1987 Journal of combinatorial theory. Series B (Print)  
The size Ramsey number for local kcolorings, ik,(G), can be defined analogously as mini IE(H)I : HE L@&(G) >. We may ask about the relation of these numbers.  ...  The size Ramsey number F'(G) of a graph G was introduced in [8] . Using the notion of minimal Ramsey graphs (see Section 5 ); ?(G) = min{ IE(H)J: HE&(G)}.  ...  RAMSEY NUMBERS AND SIZE RAMSEY NUMBERS We denote by rk(G) the Ramsey number of a graph G, i.e., the minimum III for which K,n contains a monochromatic copy of G in every k-coloring of the edges of K,.  ... 
doi:10.1016/0095-8956(87)90017-7 fatcat:nep3mrrrybehdjmxahrkxdr5vm

Page 1493 of Mathematical Reviews Vol. , Issue 2001C [page]

2001 Mathematical Reviews  
No graph G is known for which local and mean k-Ramsey numbers are different.  ...  The local k-Ramsey {mean k-Ramsey] number of a graph G is the smallest order of a complete graph for which every local [mean] k-coloring ensures a monochromatic copy of G.  ... 

Large cliques and independent sets all over the place [article]

N. Alon, M. Bucić, B. Sudakov
2020 arXiv   pre-print
Over all n-vertex graphs G what is the smallest possible value of m for which any m vertices of G contain both a clique and an independent set of size log n?  ...  Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets contain both cliques and independent sets of size log n in any small subset  ...  We would like to thank David Conlon for useful conversations and remarks and anonymous referees for their comments.  ... 
arXiv:2004.04718v2 fatcat:wqbdtvpaendmpnmyfzoo6ryzku

Page 5781 of Mathematical Reviews Vol. , Issue 95j [page]

1995 Mathematical Reviews  
Summary: “In this note we obtain some lower bounds for the Ramsey numbers r;(/;r), where r;(/;r) is the least positive integer n such that for every coloring of the k-element subsets of an n- element set  ...  Radziszowski (1-RIT-C; Rochester, NY) 95j:05137 05C55 Huang, Guo Tai Some generalized Ramsey numbers. (Chinese summary) J. Math. Res. Exposition 14 (1994), no. 2, 169-174.  ... 

On the Ramsey Numbers for Bipartite Multigraphs [article]

Ming-Yang Chen, Hsueh-I. Lu, Hsu-Chun Yen
2003 arXiv   pre-print
Note that the number of colors found in a graph under m-local coloring may exceed m.  ...  Unlike the conventional m-coloring scheme in Ramsey theory which imposes a constraint (i.e., m) on the total number of colors allowed in a graph, we introduce a relaxed version called m-local coloring  ...  Acknowledgments We thank Gerard Jennhwa Chang for very helpful comments on a preliminary version of this paper.  ... 
arXiv:cs/0305006v1 fatcat:olkho6wkwrcy5p6a6l2n32ee6q

Generalized local colorings of graphs

Miroslaw Truszczyński
1992 Journal of combinatorial theory. Series B (Print)  
We prove the Ramsey property for such colorings, establish conditions for the density property and the bipartite version of the Ramsey theorem to hold, and prove the induced variant of the Ramsey theorem  ...  A local (H, k)-coloring of a graph G is a coloring of the edges of G such that edges of no subgraph of G isomorphic to a subgraph of H are colored with more than k colors.  ...  In addition, for many graphs G the usual Ramsey number rk(G) and the local Ramsey number r:,,(G) are equal or differ by 1.  ... 
doi:10.1016/0095-8956(92)90049-4 fatcat:scavypuyd5dppafxf6hxjpwiyy

Page 2472 of Mathematical Reviews Vol. , Issue 89E [page]

1989 Mathematical Reviews  
The local Ramsey number R = LR(H;,m) is defined as the minimal number R > k such that for every Gr, Gr is not local for H, or clGr > m (clG denotes the size of the maximum complete subgraph of G).  ...  For natural numbers k,n such that k <n, let G,, H, be graphs with n and k vertices, respectively; G, is said to be local for H, if for each set S of k vertices of G, the corresponding subgraph Gs is isomorphic  ... 

Local and Mean Ramsey Numbers for Trees

B. Bollobás, A. Kostochka, R.H. Schelp
2000 Journal of combinatorial theory. Series B (Print)  
It is expected that the k-local and k-mean Ramsey numbers are identical for most graphs. In fact, it would be interesting to determine whether this pair of numbers can ever differ for some graph G.  ...  K It is likely that R(T, k loc)=R(T, k mean) for all trees T and for all k. In fact, we believe that k-local and k-mean Ramsey numbers are the same for sparse graphs of large order.  ... 
doi:10.1006/jctb.2000.1950 fatcat:7rjdq6u42fgd3f5rznkgogjl3q

New bounds for Ramsey numbers R(K_k-e,K_l-e) [article]

Jan Goedgebeur, Steven Van Overberghe
2021 arXiv   pre-print
Let R(H_1,H_2) denote the Ramsey number for the graphs H_1, H_2, and let J_k be K_k-e.  ...  We present algorithms which enumerate all circulant and block-circulant Ramsey graphs for different types of graphs, thereby obtaining several new lower bounds on Ramsey numbers including: 49 ≤ R(K_3,J  ...  Acknowledgements We would like to thank Gunnar Brinkmann and Stanis law Radziszowski for useful suggestions.  ... 
arXiv:2107.04460v1 fatcat:k54a7zz2xnaqbnpszuauwuo4iq

Page 4752 of Mathematical Reviews Vol. , Issue 87i [page]

1987 Mathematical Reviews  
The author announces some new sufficient conditions for Hamiltonian cycles in terms of degrees, independence number, and toughness. {For the entire collection see MR 87g:00016.}  ...  Sheehan and the reviewer [“The Ramsey number of Ks —e”, J. Graph Theory, to appear].} {For the entire collection see MR 86g:05026.} Heiko Harborth (Braunschweig) Caro, Y. (IL-TLAV); Krasikov, I.  ... 

New Lower Bounds for 28 Classical Ramsey Numbers [article]

Geoffrey Exoo, Milos Tatarevic
2015 arXiv   pre-print
We establish new lower bounds for 28 classical two and three color Ramsey numbers, and describe the heuristic search procedures we used.  ...  Some of the other new constructions in the paper are derived from two well-known colorings: the Paley coloring of K_101 and the cubic coloring of K_127.  ...  We noticed that some of the bounds for Ramsey numbers of the form R(3, 3, k) were close to the corresponding bounds for numbers of the form R(5, k).  ... 
arXiv:1504.02403v3 fatcat:pcso7jbfqvhv5lgjir4zuk7cgy

Page 1392 of Mathematical Reviews Vol. , Issue 90C [page]

1990 Mathematical Reviews  
Parthasarathy (6-IITM) 90c:05152 05C55 Thomason, Andrew (4-CAMB) An upper bound for some Ramsey numbers. J. Graph Theory 12 (1988), no. 4, 509-517.  ...  Lower bounds for the Ramsey number r(k,k) are dealt with in some detail, including the initial result of Erdés that r(k,k) > 2*/2 for k > 3, and the series of minor improvements that followed.  ... 

Page 2925 of Mathematical Reviews Vol. , Issue 80H [page]

1980 Mathematical Reviews  
As a way of measuring minimality for mem- bers of C, we define the size Ramsey number by /(G,,G,)= min|E(G)| for G EC.  ...  This paper concerns the Ramsey numbers (B,,,C,) for books versus cycles, and relates certain of these numbers to finite projective planes, to Moore graphs, and to other interesting structures.  ... 
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