Inspired by the definition of Gowers norms we study integrals of products of multi-variate functions. The $L_p$ norms, certain trace norms, and the Gowers norms are all defined by taking the proper root of one of these integrals. These integrals are important from a combinatorial point of view as inequalities between them are useful in understanding the relation between various subgraph densities. Lov\'asz asked the following questions: (1) Which integrals correspond to norm functions? (2) What are the common properties of the corresponding normed spaces? We address these two questions. We show that such a formula is a norm if and only if it satisfies a H\"older type inequality. This condition turns out to be very useful: First we apply it to prove various necessary conditions on the structure of the integrals which correspond to norm functions. We also apply the condition to an important conjecture of Erd\H{o}s, Simonovits, and Sidorenko. Roughly speaking, the conjecture says that among all graphs with the same edge density, random graphs contain the least number of copies of every bipartite graph. This had been verified previously for trees, the $3$-dimensional cube, and a few other families of bipartite graphs. The special case of the conjecture for paths, one of the simplest families of bipartite graphs, is equivalent to the Blakley-Roy theorem in linear algebra. Our results verify the conjecture for certain graphs including all hypercubes, one of the important classes of bipartite graphs, and thus generalize a result of Erd\H{o}s and Simonovits. In fact, for hypercubes we can prove statements that are surprisingly stronger than the assertion of the conjecture. To address the second question of Lov\'asz we study these normed spaces from a geometric point of view, and determine their moduli of smoothness and convexity. These two parameters are among the most important invariants in Banach space theory. Our result in particular determines the moduli of smoothness and convexity of Gowers norms. In some cases we are able to prove the Hanner inequality, one of the strongest inequalities related to the concept of smoothness and convexity. We also prove a complex interpolation theorem for these normed spaces, and use this and the Hanner inequality to obtain various optimum results in terms of the constants involved in the definition of moduli of smoothness and convexity.

We study a type of calculus for proving inequalities between subgraph densities which is based on Jensen's inequality for the logarithmic function. As a demonstration of the method we verify the conjecture of Erdos-Simonovits and Sidorenko for new families of graphs. In particular we give a short analytic proof for a result by Conlon, Fox and Sudakov. Using this, we prove the forcing conjecture for bipartite graphs in which one vertex is complete to the other side.

We study a type of calculus for proving inequalities between subgraph densities which is based on Jensen's inequality for the logarithmic function. As a demonstration of the method we verify the conjecture of Erdos-Simonovits and Sidorenko for new families of graphs. In particular we give a short analytic proof for a result by Conlon, Fox and Sudakov. Using this, we prove the forcing conjecture for bipartite graphs in which one vertex is complete to the other side.

Inspired by the definition of Gowers norms we study integrals of products of multi-variate functions. The $L_p$ norms, certain trace norms, and the Gowers norms are all defined by taking the proper root of one of these integrals. These integrals are important from a combinatorial point of view as inequalities between them are useful in understanding the relation between various subgraph densities. Lov\'asz asked the following questions: (1) Which integrals correspond to norm functions? (2) What are the common properties of the corresponding normed spaces? We address these two questions. We show that such a formula is a norm if and only if it satisfies a H\"older type inequality. This condition turns out to be very useful: First we apply it to prove various necessary conditions on the structure of the integrals which correspond to norm functions. We also apply the condition to an important conjecture of Erd\H{o}s, Simonovits, and Sidorenko. Roughly speaking, the conjecture says that among all graphs with the same edge density, random graphs contain the least number of copies of every bipartite graph. This had been verified previously for trees, the $3$-dimensional cube, and a few other families of bipartite graphs. The special case of the conjecture for paths, one of the simplest families of bipartite graphs, is equivalent to the Blakley-Roy theorem in linear algebra. Our results verify the conjecture for certain graphs including all hypercubes, one of the important classes of bipartite graphs, and thus generalize a result of Erd\H{o}s and Simonovits. In fact, for hypercubes we can prove statements that are surprisingly stronger than the assertion of the conjecture. To address the second question of Lov\'asz we study these normed spaces from a geometric point of view, and determine their moduli of smoothness and convexity. These two parameters are among the most important invariants in Banach space theory. Our result in particular determines the moduli of smoothness and convexity of Gowers norms. In some cases we are able to prove the Hanner inequality, one of the strongest inequalities related to the concept of smoothness and convexity. We also prove a complex interpolation theorem for these normed spaces, and use this and the Hanner inequality to obtain various optimum results in terms of the constants involved in the definition of moduli of smoothness and convexity.

The central result presented in this thesis is van der Waerden's theorem on arithmetic progressions. Van der Waerden's theorem guarantees that for any integers k and r, there is an n so that however the set {1, 2, ... , n} is split into r disjoint partition classes, at least one partition class will contain a k-term arithmetic progression. Presented here are a number of variations and generalizations of van der Waerden's theorem that utilize a wide range of techniques from areas of mathematics including combinatorics, number theory, algebra, and topology. / May 2007

The study of combinatorics on words dates back at least to the beginning of the 20th century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) xx. This result was eventually used to solve some longstanding open problems in algebra and has remarkable connections to other areas of mathematics and computer science as well. This thesis will consider several different generalizations of Thue's work. In particular we shall study the properties of infinite words avoiding various types of repetitions. In Chapter 1 we introduce the theory of combinatorics on words. We present the basic definitions and give an historical survey of the area. In Chapter 2 we consider the work of Thue in more detail. We present various well-known properties of the Thue-Morse word and give some generalizations. We examine Fife's characterization of the infinite overlap-free words and give a simpler proof of this result. We also present some applications to transcendental number theory, generalizing a classical result of Mahler. In Chapter 3 we generalize a result of Seebold by showing that the only infinite 7/3-power-free binary words that can be obtained by iterating a morphism are the Thue-Morse word and its complement. In Chapter 4 we continue our study of overlap-free and 7/3-power-free words. We discuss the squares that can appear as subwords of these words. We also show that it is possible to construct infinite 7/3-power-free binary words containing infinitely many overlaps. In Chapter 5 we consider certain questions of language theory. In particular, we examine the context-freeness of the set of words containing overlaps. We show that over a three-letter alphabet, this set is not context-free, and over a two-letter alphabet, we show that this set cannot be unambiguously context-free. In Chapter 6 we construct infinite words over a four-letter alphabet that avoid squares in any arithmetic progression of odd difference. Our constructions are based on properties of the paperfolding words. We use these infinite words to construct non-repetitive tilings of the integer lattice. In Chapter 7 we consider approximate squares rather than squares. We give constructions of infinite words that avoid such approximate squares. In Chapter 8 we conclude the work and present some open problems.

Informally, a rooted map is a topologically pointed embedding of a graph in a surface. This thesis examines two problems in the enumerative theory of rooted maps. The b-Conjecture, due to Goulden and Jackson, predicts that structural similarities between the generating series for rooted orientable maps with respect to vertex-degree sequence, face-degree sequence, and number of edges, and the corresponding generating series for rooted locally orientable maps, can be explained by a unified enumerative theory. Both series specialize M(x,y,z;b), a series defined algebraically in terms of Jack symmetric functions, and the unified theory should be based on the existence of an appropriate integer valued invariant of rooted maps with respect to which M(x,y,z;b) is the generating series for locally orientable maps. The conjectured invariant should take the value zero when evaluated on orientable maps, and should take positive values when evaluated on non-orientable maps, but since it must also depend on rooting, it cannot be directly related to genus. A new family of candidate invariants, η, is described recursively in terms of root-edge deletion. Both the generating series for rooted maps with respect to η and an appropriate specialization of M satisfy the same differential equation with a unique solution. This shows that η gives the appropriate enumerative theory when vertex degrees are ignored, which is precisely the setting required by Goulden, Harer, and Jackson for an application to algebraic geometry. A functional equation satisfied by M and the existence of a bijection between rooted maps on the torus and a restricted set of rooted maps on the Klein bottle show that η has additional structural properties that are required of the conjectured invariant. The q-Conjecture, due to Jackson and Visentin, posits a natural combinatorial explanation, for a functional relationship between a generating series for rooted orientable maps and the corresponding generating series for 4-regular rooted orientable maps. The explanation should take the form of a bijection, ϕ, between appropriately decorated rooted orientable maps and 4-regular rooted orientable maps, and its restriction to undecorated maps is expected to be related to the medial construction. Previous attempts to identify ϕ have suffered from the fact that the existing derivations of the functional relationship involve inherently non-combinatorial steps, but the techniques used to analyze η suggest the possibility of a new derivation of the relationship that may be more suitable to combinatorial analysis. An examination of automorphisms that must be induced by ϕ gives evidence for a refinement of the functional relationship, and this leads to a more combinatorially refined conjecture. The refined conjecture is then reformulated algebraically so that its predictions can be tested numerically.

The pfaffian is a classical tool which can be regarded as a generalization of the determinant. The hyperpfaffian, which was introduced by Barvinok, generalizes the pfaffian to higher dimension. This was further developed by Luque, Thibon and Abdesselam. There are several non-equivalent definitions for the hyperpfaffian, which are discussed in the introduction of this thesis. Following this we examine the extension of the Matrix-Tree theorem to the Hyperpfaffian-Cactus theorem by Abdesselam, proving it without the use of the Grassman-Berezin Calculus and with the new terminology of the non-uniform hyperpfaffian. Next we look at the extension of pfaffian orientations for counting matchings on graphs to hyperpfaffian orientations for counting matchings on hypergraphs. Finally pfaffian rings and ideal s are extended to hyperpfaffian rings and ideals, but we show that under reason able assumptions the algebra with straightening law structure of these rings cannot be extended.

The study of combinatorics on words dates back at least to the beginning of the 20th century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) <i>xx</i>. This result was eventually used to solve some longstanding open problems in algebra and has remarkable connections to other areas of mathematics and computer science as well. In this thesis we primarily study several variations of the problems studied by Thue in his work on repetitions in words, including some recent connections to other areas, such as graph theory. In Chapter 1 we give a brief introduction to the subject of combinatorics on words. In Chapter 2 we use uniform morphisms to construct an infinite binary word that contains no cubes <i>xxx</i> and no squares <i>yy</i> with |<i>y</i>| &#8805; 4, thus giving a simpler construction than that of Dekking. We also use uniform morphisms to construct an infinite binary word avoiding all squares except 0², 1², and (01)², thus giving a simpler construction than that of Fraenkel and Simpson. We give some new enumeration results for these avoidance properties and solve an open problem of Prodinger and Urbanek regarding the perfect shuffle of infinite binary words that avoid arbitrarily large squares. In Chapter 3 we examine ternary squarefree words in more detail, and in Chapter 4 we study words <i>w</i> satisfying the property that for any sufficiently long subword <i>w'</i> of <i>w</i>, <i>w</i> does not contain the reversal of <i>w'</i> as a subword. In Chapter 5 we discuss an application of the property of squarefreeness to colourings of graphs. In Chapter 6 we study strictly increasing sequences (<i>a</i>(<i>n</i>))<i>n</i>&#8805;0 of non-negative integers satisfying the equation <i>a</i>(<i>a</i>(<i>n</i>)) = <i>dn</i>. Finally, in Chapter 7 we give a brief conclusion and present some open problems.

Coin-flipping is the cryptographic task of generating a random coin-flip between two mistrustful parties. Kitaev discovered that the security of quantum coin-flipping protocols can be analyzed using semidefinite programming. This lead to his result that one party can force a desired coin-flip outcome with probability at least 1/√2. We give sufficient background in quantum computing and semidefinite programming to understand Kitaev's semidefinite programming formulation for coin-flipping cheating strategies. These ideas are specialized to a specific class of protocols singled out by Nayak and Shor. We also use semidefinite programming to solve for the maximum cheating probability of a particular protocol which has the best known security. Furthermore, we present a family of protocols where one party has a greater probability of forcing an outcome of 0 than an outcome of 1. We also discuss a computer search to find specific protocols which minimize the maximum cheating probability.