Abstract. Szemerédi’s Regularity Lemma proved to be a very powerful tool in extremal graph theory with a large number of applications. Chung [Regularity lemmas for hypergraphs and quasi-randomness, Random

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.

A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most well-studied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with one-sided error, and with query complexity depending only on ɛ. This result unifies several previous results in the area of property testing, and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemerédi’s Regularity Lemma. The main ideas behind this application may be useful in characterizing all testable graph properties, and in generally studying graph property testing. As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph-theory, is that for any monotone graph property P, any graph that is ɛ-far from satisfying P, contains a subgraph of size depending on ɛ only, which does not satisfy P. Finally, we prove the following compactness statement: If a graph G is ɛ-far from satisfying a (possibly infinite) set of monotone graph properties P, then it is at least δP(ɛ)-far from satisfying one of the properties.

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.

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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.

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Abstract. There has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has been the sequence of Gowers uniformity norms U d (G), d = 1, 2, 3,... on a finite additive group G; in particular, to detect arithmetic progressions of length k in G it is important to know under what circumstances the U k−1 (G) norm can be large. The U 1 (G) norm is trivial, and the U 2 (G) norm can be easily described in terms of the Fourier transform. In this paper we systematically study the U 3 (G) norm, defined for any function f: G → C on a finite additive group G by the formula

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