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.

We study the following one-person game against a random graph process: the Player's goal is to 2-colour a random sequence of edges of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edge.

While it is not hard to prove that the expected length of this game is about , the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in. Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with edges, where c is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round.

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.

A finite or infinite matrix A with rational entries is called partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic vector x with . Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular.

While in the finite case partition regularity is well understood, very little is known in the infinite case. Our aim in this paper is to present some of the natural and appealing open problems in the area.

In a previous paper we found a relation between the ranks of co-diagonal matrices (matrices with zeroes in their diagonal and nonzeroes elsewhere) and the quality of explicit Ramsey graph constructions. We also gave a construction based on the BBR polynomial of Barrington, Beigel and Rudich. In the present work we give another construction for low-rank co-diagonal matrices, based on a modular sieve formula.

We consider analogues of van der Waerden's theorem and Szemerédi's theorem, where arithmetic progressions are replaced by binary trees with a fixed distance between successive vertices. The proofs are based on some novel recurrence properties for Markov processes.

Given a positive integer n and a family of graphs, let denote the maximum number of colours in an edge-colouring of such that no subgraph of belonging to has distinct colours on its edges. Erdös, Simonovits and Sós [6] conjectured for fixed k with that . This has been proved for . For general k, in this paper we improve the previous bound of to . For even k, we further improve it to . We also prove that , which is sharp.

We show that the constant colourings and the one-to-one colourings are insufficient for a canonical version of a certain theorem in Ramsey theory.

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.

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.

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

A subgraph H in an edge-colouring is properly coloured if incident edges of H are assigned different colours, and H is rainbow if no two edges of H are assigned the same colour. We study properly coloured subgraphs and rainbow subgraphs forced in edge-colourings of complete graphs in which each vertex is incident to a large number of colours.

It is proved that there are infinitely many integers n such that n and n + 1 have the same number of distinct prime divisors.

We show that, if f : M2×2 → R is rank-one convex on the hyperboloid is the set of 2 × 2 real symmetric matrices, then f can be approximated by quasi-convex functions on M2×2 uniformly on compact subsets of . Equivalently, every gradient Young measure supported on a compact subset of is a laminate.

Let R be an integral domain with quotient field K and let X be an indeterminate. A result of W. C. Waterhouse states that, if each quadratic polynomial f ∈ R[X] which factors into linear polynomials in K[X] also factors into linear polynomials in R[X], then every irreducible element in R is prime. In this note the rings which satisfy the hypothesis of this theorem are characterized, and compared to the rings for which each polynomial f ∈ R[X] which factors into two polynomials of positive degree in K[X] also factors into two polynomials of positive degree in R[X]. Relevant examples are furnished via the pullback construction.

A generalization of the following two facts is proved: the class of generalized zonoids is dense in the class of convex compact bodies with a centre of symmetry, and the class of generalized triangle bodies is dense in the class of all convex compact bodies (proved by Schneider). The main result is deduced from a more general representation theoretical statement.