We prove that the chromatic Ramsey number of every odd wheel W2k+1, k ≥ 2 is 14. That is, for every odd wheel W2k+1, there exists a 14-chromatic graph F such that when the edges of F are two-coloured, there is a monochromatic copy of W2k+1 in F , and no graph F with chromatic number 13 has the same property. We ask whether a natural extension of odd wheels to the family of generalized Mycielski...

Let C (3) n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n...

For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G2. Let Cm denote a cycle of length m and Kn a complete graph of order n. We show that R(C8, K8) = 50. c © 2007 Elsevier B.V. All rights reserved.

Here we prove a stability version of a Ramsey-type Theorem for paths. Thus in any 2-coloring of the edges of the complete graph Kn we can either find a monochromatic path substantially longer than 2n/3, or the coloring is close to the extremal coloring.

We determine the 2-color Ramsey number of a connected triangle matching c(nK3) which is a graph with n vertex disjoint triangles plus (at least n − 1) further edges that keep these triangles connected. We obtain that R(c(nK3), c(nK3)) = 7n − 2, somewhat larger than in the classical result of Burr, Erdős and Spencer for a triangle matching, R(nK3, nK3) = 5n. The motivation is to determine the Ra...

We prove that for every ε > 0 there are α > 0 and n0 ∈ N such that for all n ≥ n0 the following holds. For any two-colouring of the edges of Kn,n,n one colour contains copies of all trees T of order k ≤ (3− ε)n/2 and with maximum degree ∆(T ) ≤ n. This answers a conjecture of Schelp.

The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi–coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition...

Let Q(n, χ) denote the minimum clique size an n-vertex graph can have if its chromatic number is χ . Using Ramsey graphs we give an exact, albeit implicit, formula for the case χ ≥ (n + 3)/2.

Suppose that T is an acyclic r-uniform hypergraph, with r ≥ 2. We define the (t-color) chromatic Ramsey number χ(T, t) as the smallest m with the following property: if the edges of any m-chromatic r-uniform hypergraph are colored with t colors in any manner, there is a monochromatic copy of T . We observe that χ(T, t) is well defined and ⌈ R(T, t)− 1 r − 1 ⌉ + 1 ≤ χ(T, t) ≤ |E(T )| + 1 where R...

A k-matching in a hypergraph is a set of k edges such that no two of these edges intersect. The anti-Ramsey number of a k-matching in a complete s-uniform hypergraph H on n vertices, denoted by ar(n, s, k), is the smallest integer c such that in any coloring of the edges of H with exactly c colors, there is a k-matching whose edges have distinct colors. The Turán number, denoted by ex(n, s, k),...