Szemer'edi's Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by random-looking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. Recently quite a few new results were obtained by using the Regularity Lemma, and also some new variants and generalizations appeared. In this survey we describe some typical applications and some generalizations. Contents Preface 1. Introduction 2. How to apply the Regularity Lemma 3. Early applications 4. Building large subgraphs 5. Embedding trees 6. Bounded degree spanning subgraphs 7. Weakening the Regularity Lemma 8. Strengthening the Regularity Lemma 9. Algorithmic questions 10. Regularity and randomness Preface Szemer'edi's Regularity Lemma [121] is one of the most powerful tools of (extremal) graph theory. It was invented as an auxiliary lemma in the proof of the famous conjectu...

Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.

The classical Poincar6 recurrence theorem asserts that under the action of a measure preserving transformation T of a finite measure space (X, ~, p.), every set A of positive measure recurs in the sense that for some n> 0,/z (T-'A n A)> 0. In [1] this was extended to multiple recurrence: the transformations T, T2,..., T k have a common power satisfying /x (A n T-hA n... n T-k"A)> 0 for a set A of positive measure. We also showed that this result implies Szemer6di's theorem stating that any set of integers of positive upper density contains arbitrarily long arithmetic progressions. In [2] a topological analogue of this is proved: if T is a homeomorphism of a compact metric space X, for any e>0 and k = 1,2,3,..-, there is a point x E X and a common power of T, T 2, 9 9 9 T k such that d(x, Tnx) < e, d(x, T2"x) < e,. 9 d(x, Tk~x) < e. This (weaker) result, in turn, implies van der Waerden's theorem on arithmetic progressions for partitions of the integers. Now in this case a virtually identical argument shows that the topological result is true for any k commuting transformations. This would lead one to expect that the measure theoretic result is also true for arbitrary commuting transformations. (It is easy to give a counterexample with noncommuting transformations.) We prove this in what follows. Theorem A. Let (X,~,/z) be a measure space with /z(X)<oo, let T~, T2, 9 9 9 Tk be commuting measure preserving transformations of X and let A E B with tz (A)> O. Then lim inf 1 N ~-N 1 Iz ( T~'A A Tj~A N...A T~A)>O. A corollary is the multidimensional extension of Szemer6di's theorem: Theorem B. Let S C Z " be a subset with positive upper density and let F C Z " be any finite configuration. Then there exists an integer d and a vector n E Z " such that n+dFCS.

Abstract. Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓ-partite graph with V (G) = V1 ∪ · · · ∪ Vℓ and |Vi | = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are ε-regular of density d for ℓ 1 ≤ i < j ≤ ℓ and ε ≪ d, then G contains (1 ± fℓ(ε))d 2 × nℓ cliques Kℓ, where fℓ(ε) → 0 as ε → 0.

This paper shows for a bounded tile that all tilings it gives of R are periodic, and that there are finitely many translation-equivalence classes of such tilings. The main result of the paper is that for any tiling of R by a bounded tile, any two tiles in the tiling differ by a rational multiple of the minimal period of the tiling. This result implies a structure theorem characterizing such tiles in terms of complementing sets for finite cyclic groups. 1. Introduction

In this paper the following improvement on Freiman’s theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A ⊂ Z be a finite set such that |A + A | < α|A|. Then A is contained in a proper d-dimensional progression P, where d ≤ [α − 1] and log(|P|/|A|) < Cα 2 (log α) 3. Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa’s method, which we improve in several places, as well as Y. Bilu’s proof of Freiman’s theorem.

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.

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 prove that if a countable discrete group Γ is w-rigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. Γ = SL(2, Z) ⋉ Z 2, or Γ = H × H ′ with H an infinite Kazhdan group and H ′ arbitrary), and V is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. V countable discrete, or separable compact), then any V-valued measurable cocycle for a measure preserving action Γ � X of Γ on a probability space (X, µ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action Γ � [0,1] Γ) is cohomologous to a group morphism of Γ into V. We use the case V discrete of this result to prove that if in addition Γ has no non-trivial finite normal subgroups then any orbit equivalence between Γ � X and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ ≃ Λ. There has recently been increasing interest in the study of measure preserving actions of groups on (non-atomic) probability spaces up to orbit equivalence (OE), i.e. up to isomorphisms of probability spaces taking the orbits of one action onto the orbits of

Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m),..., x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties. Contents