IMPA Master's Thesis Low-Complexity Decompositions of Combinatorial Objects Author: Supervisor: Davi Castro-Silva Roberto Imbuzeiro Oliveira A thesis submitted in fulfillment of the requirements for the degree of Master in Mathematics at Instituto de Matem´aticaPura e Aplicada April 15, 2018 iii Contents 1 Introduction 1 1.1 High-level overview of the framework.........................1 1.2 Remarks on notation and terminology........................2 1.3 Examples and structure of the thesis.........................3 2 Abstract decomposition theorems7 2.1 Probabilistic setting..................................7 2.2 Structure and pseudorandomness...........................8 2.3 Weak decompositions..................................9 2.3.1 Weak Regularity Lemma........................... 10 2.4 Strong decompositions................................. 10 2.4.1 Szemer´ediRegularity Lemma......................... 11 3 Counting subgraphs and the Graph Removal Lemma 15 3.1 Counting subgraphs globally.............................. 15 3.2 Counting subgraphs locally and the Removal Lemma................ 17 3.3 Application to property testing............................ 19 4 Extensions of graph regularity 21 4.1 Regular approximation................................. 21 4.2 Relative regularity................................... 23 5 Hypergraph regularity 27 5.1 Intuition and definitions................................ 27 5.2 Regularity at a single level............................... 28 5.3 Regularizing all levels simultaneously......................... 30 6 Dealing with sparsity: transference principles 33 6.1 Subsets of pseudorandom sets............................. 33 6.2 Upper-regular functions................................ 34 6.3 Green-Tao-Ziegler Dense Model Theorem...................... 37 7 Transference results for L1 structure 41 7.1 Relationships between cut norm and L1 norm.................... 41 7.2 Inheritance of structure lemmas............................ 43 7.3 A \coarse" structural correspondence......................... 45 7.4 A “fine" structural correspondence.......................... 46 7.4.1 Proof of Theorem 7.2............................. 47 8 Extensions and open problems 51 Bibliography 53 1 Chapter 1 Introduction Many times in Mathematics and Computer Science we are dealing with a large and general class of objects of a certain kind and we wish to obtain non-trivial results which are valid for all objects belonging to this class. This may be a very hard task if the possible spectrum of behavior for the members of this class is very broad, since it is unlikely that any single argument will hold uniformly along this whole spectrum. Such results may be easy (or easier) to obtain when the class we are dealing with is highly structured, in the sense that one can encode its elements in such a way that the description of each object has a relatively small size; then it may be possible to use this structure to prove results valid uniformly over all objects in this class, or to do a case-by-case analysis to obtain such results. At the other end of the spectrum there are the random objects, which have a very high complexity in the sense that any description of a randomly chosen object must specify the random choices made at each point separately, and thus be very large if the object in consideration is large. However, for such objects there are various \concentration inequalities" which may be used to obtain results valid with high probability over the set of random choices made. Therefore, if we can decompose every object belonging to the general class we are interested in into a \highly structured" component (which has low complexity) and a \pseudorandom" component (which mimics the behavior of random objects in certain key statistics), then we may analyze each of these components separately by different means and so be able to obtain results which are valid for all such objects. An illustrative example of a \structure-pseudorandomness" decomposition of this kind is Szemer´edi'scelebrated Regularity Lemma [31]. This important result roughly asserts that the vertices of any graph G may be partitioned into a bounded number of equal-sized parts, in such a way that for almost all pairs of partition classes the bipartite graph between them is random-like. Both the upper bound we get for the order of this partition and the quality of the pseudorandomness behavior of the edges between these pairs depend only on an accuracy parameter " we are at liberty to choose. In this example, the object to be decomposed is the edge set E of a given arbitrary graph G = (V; E), which belongs to the \general class" of all graphs. The structured component then represents the pairs (Vi;Vj) of partition classes together with the density of edges between them, and it has low complexity because the order of the partition is uniformly bounded for all graphs. The pseudorandom component represents the actual edges between these pairs, and has a random-like property known as "-regularity which we will define in the next chapter. This result has many applications in Combinatorics and Computer Science (see, e.g., [20, 21] for a survey), and it has inspired numerous other decomposition results in a similar spirit both inside and outside Graph Theory. In this work we aim to survey many decomposition theorems of this form present in the literature. We provide a unified framework for proving them and present some new results along these same lines. 1.1 High-level overview of the framework In our setting, the combinatorial objects to be decomposed will be represented as functions defined over a discrete space X. This identification does not give much loss in generality, since given a combinatorial object O (such as a graph, hypergraph or additive group), we may usually 2 Chapter 1. Introduction identify some underlying discrete space X for this kind of object and then represent O as a function fO defined on X. We endow X with a probability measure P, so that the objects considered may be viewed as random variables, and define a family C of \low-complexity" subsets of X. The specifics of both the probability measure P and the structured family C will depend on the application at hand, and it is from them that we will define our notions of complexity and pseudorandomness. The sets belonging to C are seen as the basic structured sets, which have complexity 1, and any subset of X which may be obtained by boolean operations from at most k of these basic structured sets A1; ··· ;Ak 2 C is said to have complexity at most k according to C. We then say two functions g; h : X ! R are "-indistinguishable according to C if, for all sets A 2 C, we have that jE [(g − h) 1A]j ≤ ". Intuitively, this means that we are not able to effectively distinguish between h and g by taking their empirical averages over random elements chosen from one of the basic sets in C. A function f : X ! R is then said to be "-pseudorandom if it is "-indistinguishable from the constant function 1 on X. Thus pseudorandom functions are in some sense uniformly distributed over structured sets, mimicking random functions of mean 1 defined on X. These concepts are closely related to the notions of pseudorandomness and indistinguishability in the area of Computational Complexity Theory (in the non-uniform setting). In this case, we have a collection F of “efficiently computable" boolean functions f : X ! f0; 1g (which are though of as adversaries), and two distributions A and B on X are said to be "-indistinguishable by F if jP (f(A) = 1) − P (f(B) = 1)j ≤ " 8f 2 F A distribution R is then said to be "-pseudorandom for F if it is "-indistinguishable from the uniform distribution UX on X. Intuitively, this means that no adversary from the class F is able to distinguish R from UX with non-negligible advantage. This is completely equivalent to our definitions, if we identify each function f in F with its support f −1(1) in X, and identify the distributions A, B with the functions g(x) := P(A = x)·jXj, h(x) := P(B = x) · jXj. Then jP (f(A) = 1) − P (f(B) = 1)j = E (g − h) 1f −1(1) ; where the expectation on the right-hand side is with respect to the uniform distribution. In our abstract decomposition theorems given in Chapter2, it will be convenient to deal with σ-algebras on X rather than with subsets of X; since a σ-algebra on a finite space X is a finite collection of subsets of X, the intuition will be essentially the same. However, this change will make it simpler to apply tools such as the Cauchy-Schwarz inequality and Pythagoras' theorem, which will be both very important in our energy-increment arguments. Moreover, we will also require pseudorandom functions to have no correlation in average value to the structured sets, and thus be "-indistinguishable from the zero function on X. Since the expected value function is linear, this \translation" in our definition makes no important difference. The framework as described here will be retaken in Chapter6, when we talk about transfer- ence principles and the Dense Model Theorem. 1.2 Remarks on notation and terminology We will be mainly interested in very large objects, and use the usual asymptotic notation O, Ω, and Θ with subscripts indicating parameters the implied constant is allowed to depend on. For instance, Oα,β(X) denotes a quantity bounded in absolute value by Cα,βjXj for some quantity Cα,β depending only on α and β. We write Ea2A;b2B to denote the expectation when a in chosen uniformly from the set A and b is chosen uniformly from the set B, both choices being independent. For any real numbers a and b, we write x = a ± b to denote a − b ≤ x ≤ a + b. Given an integer n ≥ 1, we write [n] for the set f1; ··· ; ng. If A is a set and k is an integer, A we write k to denote the collection of all k-element subsets of A.