We investigate Ramsey numbers of bounded degree graphs and provide an interpolation between known results on the Ramsey numbers of general bounded degree graphs and bounded degree graphs of small bandwidth. Our main theorem implies that there exists a constant c such that for every ∆, there exists β such that if G is an n-vertex graph with maximum degree at most ∆ having a homomorphism f into a...

The Ramsey degree of an ordinal α is the least number n such that any colouring of the edges of the complete graph on α using finitely many colours contains an n-chromatic clique of order type α. The Ramsey degree exists for any ordinal α < ω. We provide an explicit expression for computing the Ramsey degree given α. We further establish a version of this result for automatic structures. In thi...

As usual, for simple graphs G and H, let the Ramsey number r(G,H) be defined as the least number n such that for any graph K of order n, either G is a subgraph of K or H is a subgraph of / ( . We shall establish the values of r(aC~,bCs) and r(aCv, bC7) almost precisely (where nG is the graph consisting of n vertex disjoint copies of G) extending the work of Mizuno and Sato, who proved similar r...

For a fixed graph H on k vertices, we investigate the graphs, G, such that for any partition of the vertices of G into k color classes, there is a transversal of that partition inducing H. For every integer k ≥ 1, we find a family F of at most six graphs on k vertices such that the following holds. If H / ∈ F , then for any graph G on at least 4k − 1 vertices, there is a k-coloring of vertices ...

Let G = (V,E) be a d-regular graph of order n. Let Gp be the random subgraph of G for which each edge is selected from E(G) independently at random with probability p. For a fixed graph H, define m(H) := max{e(H ′)/(v(H ′)− 1) : H ′ ⊆ H}. We prove that n(m(H)−1)/m(H)/d is a threshold function for Gp to satisfy Ramsey, induced Ramsey, and canonical Ramsey properties with respect to vertex colori...