The Ramsey number of a graph with bounded maximum degree

We consider the numbers associated with Ramsey's theorem as it pertains to partitions of the pairs of elements of a set into two classes. Our purpose is to give a unified development of enumerative techniques which give sharp upper bounds on these numbers and to give constructive methods for partitions to determine lower bounds on these numbers. Explicit computations include the values of R(3, 6) and R(3, 7) among others. Our computational techniques yield the upper bound R(x, y) < cyX-llog log y/log y for x ~> 3.

Pacific Journal of Mathematics, 1974

Let χ(G) denote the chromatic number of a graph G. For positive integers n ίf n 2 , , n k (k ^ 1) the chromatic Ramsey number χ(n ίf n 2 , , n k) is defined as the least positive integer p such that for any factorization K p-U*=i G if χ{G t) Ξ> w* for at least one i,l ^i ^k. It is shown that x(n u n 2 , , n k) = 1 + ΓR=i (n t-1). The vertex-arboricity a(G) of a graph G is the fewest number of subsets into which the vertex set of G can be partitioned so that each subset induces an acyclic graph. For positive integers n lt n 2 , , w* (ft ^ 1) the vertexarboricity Ramsey number a(n lt n 2 ,-,n k) is defined as the least positive integer p such that for any factorization K p-U*=i Gi> d(G t) ^ Ύii for at least one i, 1 rg % <Ξ k. It is shown that a(n lt n 2 , , n k) = 1 + 2k ΓR=i (n t-1).

Applied Mathematical Sciences

The Ramsey number R(G1,G2,…,Gk) is the least integer p so that for any k-edge coloring of the complete graph Kp, there is a monochromatic copy of Gi of color i. In this paper, we derive upper bounds of R(G1,G2,…,Gk) for certain graphs Gi. In particular, these bounds show that R(9,9)⩽6588 and R(10,10)⩽23556 improving the previous best bounds of 6625 and 23854.

2010

Given a fixed integer $n$, we prove Ramsey-type theorems for the classes of all finite ordered $n$-colorable graphs, finite $n$-colorable graphs, finite ordered $n$-chromatic graphs, and finite $n$-chromatic graphs.

Discrete Mathematics, 1994

The Ramsey number r(H, G) is defined as the minimum N such that for any coloring of the edges of the N-vertex complete graph K N in red and blue, it must contain either a red H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K 3 , G) ≤ 2q + 1 where G has q edges. In other words, any graph on 2q + 1 vertices with independence number at most 2 contains every (isolate-free) graph on q edges. This establishes a 1980 conjecture of Harary. The result is best possible as a function of q.

SIAM Journal on Discrete Mathematics

For given simple graphs G 1 and G 2 , the size Ramsey numberR(G 1 , G 2) is the smallest positive integer m, where there exists a graph G with m edges such that in any edge coloring of G with two colors red and blue, there is either a red copy of G 1 or a blue copy of G 2. In 1981, Erdős and Faudree investigated the size Ramsey numberR(K n , tK 2), where K n is a complete graph on n vertices and tK 2 is a matching of size t. They obtained the value ofR(K n , tK 2) when n ≥ 4t − 1 as well as for t = 2 and asked for the behavior of these numbers when t is much larger than n. In this regard, they posed the following interesting question: For every positive integer n, is it true that lim t→∞R (K n , tK 2) tR(K n , K 2) = min n+2t−2 2 t n 2 | t ∈ N ? In this paper, we obtain the exact value ofR(K n , tK 2) for every positive integers n, t and as a byproduct, we give an affirmative answer to the question of Erdős and Faudree.

European Journal of Combinatorics, 2010

1996

Agsrnacr. The set Ramsey number r.(t, (l)) is the smallest integer r such that if the edges of a complete graph K, are 2-colored, then there will be a graph with n vertices a,nd /c edges in the first cblorbr a graph with n vertices ana (|) (e.g. a complete graph) in the second eolor. For each n ) 3 and 1 < k < n, the set Ramsey numbers r.(f, (l)) are determined. One approach to get some insight into r(K") was suggested in , where Ramsey numbers for sets of graphs with fixed numbers of vertices and edges were considered. Thus, the following definition. Definition 1. For positive integers n'L,r\s,t with 1 ( s ( (\) ana t < t < (Z) the set Ramsey number r^,n(s,t) is the smallest integer r such that for arry 2-coloring of the edges of a complete graph K,, there is either a graph with m vertices and's edges in the first color, or a graph with n vertices and t edges in the second color. When rrl : rL, r*,^(s,t) will be expressed more compactly as r*(s,t). Associated with fixed positive integers rn and n there is an (!) UV (i) array of Ramsey numbers (r*,n(t,t)) for 1 ( s ( (f) ana 1 <, < (!) ttrat represent sets of graphs with fixed numbers of edges. For small values of m and n the array of Ramsey numbers (r*,n(t,t)) have been determined. The values for m :3 and 3 I n 17 were determined in [4], and the values for m: 4 and 4 1n 15 were determined in , except ior ra,s((l), (!)) : r(Ka, K5), which has now been shown to be 25 by McKay and R^adziszowski (see [fl). All but 5 cf the values of (r5(s,t)) were determined in [5]. For Ramsey numbers of more general sets of small graphs see [2].

2021

A set of vertices X ⊆ V in a simple graph G(V, E) is irredundant if each vertex x ∈ X is either isolated in the induced subgraph X or else has a private neighbor y ∈ V \ X that is adjacent to x and to no other vertex of X. The irredundant Ramsey number s(m, n) is the smallest N such that in every red-blue coloring of the edges of the complete graph of order N , either the blue subgraph contains an m-element irredundant set or the red subgraph contains an n-element irredundant set. The mixed Ramsey number t(m, n) is the smallest N for which every red-blue coloring of the edges of K N yields an m-element irredundant set in the blue subgraph or an n-element independent set in the red subgraph. In this paper, we first improve the upper bound of t(3, n); using this result, we confirm that a conjecture proposed by Chen, Hattingh, and Rousseau, that is, lim n→∞ t(m,n) r(m,n) = 0 for each fixed m ≥ 3, is true for m ≤ 4. At last, we prove that s(3, 9) and t(3, 9) are both equal to 26.

Australasian Journal of Combinatorics, 2017

Giraud (1968) demonstrated a process for constructing cyclic Ramsey graph colourings, starting from a known cyclic 'prototype' colouring, adding edges of a single new colour, and producing a larger cyclic pattern. This paper describes an extension of that construction which allows any number of new colours to be introduced simultaneously, by using two multicolour prototypes, each of which is a linear Ramsey graph. The resulting colouring is also linear, which allows the process to be applied iteratively. It is then proved that a simple formula resulting from the new construction provides improved lower bounds for many Ramsey numbers. Giraud's recursive formula is proved for all linear cases, as a corollary. The formula resulting from the new construction is applied to produce new lower bounds for several particular Ramsey numbers, including R 5 (4) ≥ 4176, R 4 (5) ≥ 3282, R 5 (5) ≥ 33495 and R 4 (6) ≥ 20202. For some larger R r (3), the construction produces new lower bounds that improve over the construction described by Chung (1973)-including R 12 (3) ≥ 575666. The paper goes on to explore the general limits, implied by the formula , for lower bounds for the Ramsey numbers R r (k). Specific lower bounds are derived in the form lim r→∞ R r (k) 1/r ≥ g k .