Upper Bounds for the Chromatic Numbers of Euclidean Spaces with Forbidden Ramsey Sets

Mathematical Notes - The chromatic number of a Euclidean space ? n with a forbidden finite set C of points is the least number of colors required to color the points of this space so that no...     

Ramsey Numbers for Trees II

Czechoslovak Mathematical Journal - Let r(G1, G2) be the Ramsey number of the two graphs G1 and G2. For n1 ? n2 ? 1 let S(n1, n2) be the double star given by $$V\left( {S\left(...     

Induced Ramsey Numbers

   We investigate the induced Ramsey number of pairs of graphs (G, H). This number is defined to be the smallest possible order of a graph Γ with the property that, whenever its edges are coloured red and blue, either a red induced copy of G arises or else a blue induced copy of H arises. We show that, for any G and H with , we have

Poset Ramsey Numbers for Boolean Lattices

Order - For each positive integer n, let Qn denote the Boolean lattice of dimension n. For posets $P, P^{\prime }$ , define the poset Ramsey number $R(P,P^{\prime })$ to be the least N such that...     

Linear Ramsey Numbers for Bounded-Degree Hypergrahps

We show that the Ramsey number is linear for every uniform hypergraph with bounded degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. [V. Chvátal, V. Rödl, E. Szemerédi and W.T. Trotter, Jr., The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), pp. 239–243] showed in 1983. Our proof demonstrates the potential of a new regularity lemma by [Y. Ishigami, A simple regularization of hypergraphs, preprint, arXiv:math/0612838 (2006)].     

Some tree-star Ramsey Numbers

The Ramsey Number r(G₁, G₂) is the least integer N such that for every graph G with N vertices, either G has the graph G₁ as a subgraph or $\text{G}$, the complement of G, has the graph G₂ as a subgraph.In this paper we embed the paths P_m in a much larger class $\text{T}$ of trees and then show how some evaluations by T. D. Parsons of Ramsey numbers r(P_m, K_1,n), where K_1,n is the star of degree n, are also valid for r(T_m, K_1,n) where T_m ∈ $\text{T}$.     

Some Bipartite Ramsey Numbers

Consider a complete bipartite graph K(s, s) with p = 2s points. Let each line of the graph have either red or blue colour. The smallest number p of points such that K(s, s) always contains red K(m, n) or blue K(m, n) is called bipartite Ramsey number denoted by rb(K(m, n), K(m, n)). In this paper, we show that

Canonical Pattern Ramsey Numbers

A color pattern is a graph whose edges have been partitioned into color classes. A family of color patterns is a Ramsey family provided there is some sufficiently large integer N such that in any edge coloring of the complete graph K_N there is an (isomorphic) copy of at least one of the patterns from . The smallest such N is the Ramsey number of the family . The classical Canonical Ramsey theorem of Erds and Rado asserts that the family of color patterns is a Ramsey family if it consists of monochromatic, rainbow (totally multicolored) and lexically colored complete graphs. In this paper we treat the asymmetric case by studying the Ramsey number of families containing a rainbow triangle, a lexically colored complete graph and a fixed arbitrary monochromatic graph. In particular we give asymptotically tight bounds for the Ramsey number of a family consisting of rainbow and monochromatic triangle and a lexically colored K_N. Among others, we prove some canonical Ramsey results for cycles.     

若干广义Ramsey数

本文讨论了关于树对完全图删去一些相交的三阶路的广义Ｒａｍｓｅｙ数Ｒ（Ｔ_ｍ，Ｋ_n－ｔＰ₃）和关路对完全图删去一些不相交的三阶完全图的广义Ｒａｍｓｅｙ数Ｒ（Ｐ_ｍ，Ｋ_n－ｔＫ₃），获得如下结果：1．如果m≥3,n≥3，那么Ｒ（Ｔ_ｍ，Ｋ_n－ｔＰ₃）=(m-1)(n-t-1)+1,0≤t≤[n/3].２．若m≥4,n,T≥1，则Ｒ（P_ｍ，Ｋ_n－ｔK₃）=(m-1)(n+2t-1)+1.从而，这两个结果部分地回答了１９８３年Ｒ．Ｊ．Ｇｏｕｌｄ和Ｍ．Ｓ．Ｊａｃｏｂｓｏｎ在［１］中提出的未解决问题．     

On Geometric Graph Ramsey Numbers

For any two-colouring of the segments determined by 3n − 3 points in general position in the plane, either the first colour class contains a triangle, or there is a noncrossing cycle of length n in the second colour class, and this result is tight. We also give a series of more general estimates on off-diagonal geometric graph Ramsey numbers in the same spirit. Finally we investigate the existence of large noncrossing monochromatic matchings in multicoloured geometric graphs. Research partially supported by Hungarian Scientific Research Grants OTKA T043631 and NK67867.     

Three-Color Ramsey Numbers For Paths

We prove—for sufficiently large n—the following conjecture of Faudree and Schelp:

Some Bistar Bipartite Ramsey Numbers

For bipartite graphs G ₁, G ₂, . . . ,G _k , the bipartite Ramsey number b(G ₁, G ₂, . . . , G _k ) is the least positive integer b so that any colouring of the edges of K _b,b with k colours will result in a copy of G _i in the ith colour for some i. A tree of diameter three is called a bistar, and will be denoted by B(s, t), where s ≥ 2 and t ≥ 2 are the degrees of the two support vertices. In this paper we will obtain some exact values for b(B(s, t), B(s, t)) and b(B(s, s), B(s, s)). Furtermore, we will show that if k colours are used, with k ≥ 2 and s ≥ 2, then \({b_{k}(B(s, s)) \leq \lceil k(s - 1) + \sqrt{(s - 1)^{2}(k^{2} - k) - k(2s - 4)} \rceil}\) . Finally, we show that for s ≥ 3 and k ≥ 2, the Ramsey number \({r_{k}(B(s, s)) \leq \lceil 2k(s - 1)+ \frac{1}{2} + \frac{1}{2} \sqrt{(4k(s - 1) + 1)^{2} - 8k(2s^{2} - s - 2)} \rceil}\) .     

On Some Three-Color Ramsey Numbers

 In this paper we study three-color Ramsey numbers. Let K _i,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G ₁, G ₂ and G ₃, if r(G ₁, G ₂)≥s(G ₃), then r(G ₁, G ₂, G ₃)≥(r(G ₁, G ₂)−1)(χ(G ₃)−1)+s(G ₃), where s(G ₃) is the chromatic surplus of G ₃; (ii) (k+m−2)(n−1)+1≤r(K _1,k , K _1,m , K _n )≤ (k+m−1)(n−1)+1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m≥k≥2, there is a constant c such that r(K _k,m , K _k,m , K _n )≤c(n/logn), and r(C _2m , C _2m , K _n )≤c(n/logn) ^m/(m−1) for sufficiently large n. Received: July 25, 2000 Final version received: July 30, 2002 RID="*" ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of education ministry of China, and the foundations of Jiangsu Province Acknowledgments. The authors are grateful to the referee for his valuable comments. AMS 2000 MSC: 05C55     

On Ramsey Numbers for Arbitrary Sequences of Graphs

Doklady Mathematics - In this work, we study nontrivial generalizations of Ramsey numbers to the case of arbitrary sequences of graphs. For many classes of sequences, exact values or asymptotics of...     

Ramsey Numbers for Trees of Small Maximum Degree

For a tree T we write and , , for the sizes of the vertex classes of T as a bipartite graph. It is shown that for T with maximum degree , the obvious lower bound for the Ramsey number R(T,T) of is asymptotically the correct value for R(T,T). Received December 15, 1999 RID=" " ID=" " The first and third authors were partially supported by NSERC. The second author was partially supported by KBN grant 2 P03A 021 17.     

On Some Ramsey Numbers for Quadrilaterals Versus Wheels

For given graphs G ₁ and G ₂, the Ramsey number R(G ₁, G ₂) is the least integer n such that every 2-coloring of the edges of K _n contains a subgraph isomorphic to G ₁ in the first color or a subgraph isomorphic to G ₂ in the second color. Surahmat et al. proved that the Ramsey number ${R(C_4, W_n) \leq n + \lceil (n-1)/3\rceil}$ . By using asymptotic methods one can obtain the following property: ${R(C_4, W_n) \leq n + \sqrt{n}+o(1)}$ . In this paper we show that in fact ${R(C_4, W_n) \leq n + \sqrt{n-2}+1}$ for n ≥ 11. Moreover, by modification of the Erd?s-Rényi graph we obtain an exact value ${R(C_4, W_{q^2+1}) = q^2 + q + 1}$ with q ≥ 4 being a prime power. In addition, we provide exact values for Ramsey numbers R(C ₄, W _n ) for 14 ≤ n ≤ 17.