Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and Furstenberg-Katznelson [7] concerning one-dimensional and multi-dimensional arithmetic progressions respectively. In this paper we shall give a self-contained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [29] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes. 1.

In this note we discuss several combinatorial problems that can be addressed by the Regularity Method for hypergraphs. Based on recent results of Nagle, Schacht and the authors, we give here solutions to these problems. In particular, we prove the following: Let F be a k-uniform hypergraph on t vertices and suppose an n-vertex k-uniform hypergraph H contains only o(n t) copies of F. Then one can delete o(n k) edges of H to make it F-free. Similar results were recently obtained by W. T. Gowers.

A famous theorem of Szemerédi asserts that given any density 0 < δ ≤ 1 and any integer k ≥ 3, any set of integers with density δ will contain infinitely many proper arithmetic progressions of length k. For general k there are essentially four known proofs of this fact; Szemerédi’s original combinatorial proof using the Szemerédi regularity lemma and van der Waerden’s theorem, Furstenberg’s proof using ergodic theory, Gowers’ proof using Fourier analysis and the inverse theory of additive combinatorics, and Gowers’ more recent proof using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring in particular the use of transfinite induction (and thus the axiom of choice), decomposing a general ergodic system as the weakly mixing extension of a transfinite tower of compact extensions. Here we present a quantitative, self-contained version of this ergodic theory proof, and which is “elementary ” in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.

Abstract. We introduce a correspondence principle (analogous to the Furstenberg correspondence principle) that allows one to extract an infinite random graph or hypergraph from a sequence of increasingly large deterministic graphs or hypergraphs. As an application we present a new (infinitary) proof of the hypergraph removal lemma of Nagle-Schacht-Rödl-Skokan and Gowers, which does not require the hypergraph regularity lemma and requires significantly less computation. This in turn gives new proofs of several corollaries of the hypergraph removal lemma, such as Szemerédi’s theorem on arithmetic progressions. 1.

Abstract. We show that the Gaussian primes P[i] ⊆ Z[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers v0,..., vk−1, we show that there are infinitely many sets {a+rv0,..., a+rvk−1}, with a ∈ Z[i] and r ∈ Z\{0}, all of whose elements are Gaussian primes. The proof is modeled on that in [9] and requires three ingredients. The first is a hypergraph removal lemma of Gowers and Rödl-Skokan, or more precisely a slight strengthening of this lemma which can be found in [22]; this hypergraph removal lemma can be thought of as a generalization of the Szemerédi-Furstenberg-Katznelson theorem concerning multidimensional arithmetic progressions. The second ingredient is the transference argument from [9], which allows one to extend this hypergraph removal lemma to a relative version, weighted by a pseudorandom measure. The third ingredient is a Goldston-Yıldırım type analysis for the Gaussian integers, similar to that in [9], which yields a pseudorandom measure which is concentrated on Gaussian “almost primes”. 1.

Abstract. A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green-Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different. 1.

Abstract. Let Cn denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v1,..., vn and edges v1v2v3, v3v4v5, v5v6v7,..., vn−1vnv1. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of Cn, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible. 1.

The first half of this paper is mainly expository, and aims at introducing the regularity lemma of Szemerédi. Among others, we discuss an early application of the regularity lemma that relates the notions of universality and uniform distribution of edges, a form of ‘pseudorandomness’ or ‘quasi-randomness’. We then state two closely related variants of the regularity lemma for sparse graphs and present a proof for one of them. In the second half of the paper, we discuss a basic idea underlying the algorithmic version of the original regularity lemma: we discuss a ‘local’ condition on graphs that turns out to be, roughly speaking, equivalent to the regularity condition of Szemerédi. Finally, we show how the sparse version of the regularity lemma may be used to prove the equivalence of a related, local condition for regularity. This new condition turns out to give a O(n²) time algorithm for testing the quasi-randomness of an n-vertex graph.

Abstract. Szemerédi’s regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemerédi’s theorem on arithmetic progressions [19], [18]. In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a slightly stronger variant of this lemma, related to similar strengthenings of that lemma in [1]. This stronger version of the regularity lemma was extended in [21] to reprove the analogous regularity lemma for hypergraphs. 1.

Abstract. A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graph-theoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourier-analytic approach of Gowers, and the hypergraph approach of Nagle-Rödl-Schacht-Skokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different. 1.