In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these… Expand

We construct a Banach space that does not contain any infinite un- conditional basic sequence and investigate further properties of this space. For example, it has no subspace that can be written as… Expand

We prove analogues for hypergraphs of Szemeredi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional… Expand

It is shown that there exists a constant c > 0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy = z.Expand

This paper shows that the bound is necessarily of tower type, obtaining a lower bound given by a tower of 2s of height proportional to $ \log{(1/ \epsilon)} $).Expand

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost… Expand

For a certain class of algebras $\cal A$ we give a method for constructing Banach spaces $X$ such that every operator on $X$ is close to an operator in $\cal A$. This is used to produce spaces with a… Expand

A problem of Banach asks whether every infinite-dimensional Banach space which is isomorphic to all its infinite-dimensional subspaces must be isomorphic to a separable Hilbert space. In this paper… Expand