If φ is a scattered order type, and μ is a cardinal, then there exists a scattered order type ψ such that holds.

The Ramsey number, , of a graph G is the minimum integer N such that, in every 2-colouring of the edges of the complete graph on N vertices, there is a monochromatic copy of G. In 1975, Burr and Erdős posed a problem on Ramsey numbers of d-degenerate graphs, i.e., graphs in which every subgraph has a vertex of degree at most d. They conjectured that for every d there exists a constant c(d) such that for any d-degenerate graph G of order n.

In this paper we prove that for each such G. In fact, we show that, for every , sufficiently large n, and any graph H of order , either H or its complement contains a (d,n)-common graph, that is, a graph in which every set of d vertices has at least n common neighbours. It is easy to see that any (d,n)-common graph contains every d-degenerate graph G of order n. We further show that, for every constant C, there is an n and a graph H of order such that neither H nor its complement contains a -common graph.

A classical result of Sárközy states that, for any and any positive density subset E of , there exist elements x and y of E and such that . A version of this result for finite fields is derived from a recent theorem of P. Larick, a short proof of which is also given.

Given a graph Hwith no isolates, the (generalized) mixed Ramsey number is the smallest integer r such that every H-free graph of order r contains an m-element irredundant set. We consider some questions concerning the asymptotic behaviour of this number (i) with H fixed and , (ii) with m fixed and a sequence of dense graphs, in particular for the sequence . Open problems are mentioned throughout the paper.

We consider the k-party communication complexity of the problem of determining if a word w is of the form , for fixed letters . Using the well-known theorem of Hindman (a Ramsey-type result about finite subsets of natural numbers), we prove that for and 5 the communication complexity of the problem increases with the length of the word w.

Let σ be a finite relational signature, let be a set of finite complete relational structures of signature σ, and let be the countable homogeneous relational structure of signature σ which does not embed any of the structures in .

When σ consists of at most binary relations and is finite, the vertex partition behaviour of is completely analysed, in the sense that it is shown that a canonical partition exists and the size of this partition in terms of the structures in is determined. If is infinite some results are obtained, but a complete analysis is still missing.

Some general results are presented which are intended to be used in further investigations when σ contains relational symbols of arity larger than two or when the set of bounds is infinite.