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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 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.

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.

Recent work on many problems in additive combinatorics, such as Roth's Theorem, has shown the usefulness of first studying the problem in a finite field environment. Using the techniques of Bourgain to give a result in other settings such as general abelian groups, the author gives a walk through, including proof, of Roth's theorem in both the one dimensional and two dimensional cases (it would be more accurate to refer to the two dimensional case as Shkredov's Theorem). In the one dimensional case the argument is at its base Meshulam's but the structure will be essentially Green's. Let Ϝⁿ [subscript p], p ≠ 2 be the finite field of cardinality N = pⁿ. For large N, any subset A ⊂ Ϝⁿ [subscript p] of cardinality ∣A ∣≳ N ∕ log N must contain a triple of the form {x, x + d, x + 2d} for x, d ∈ Ϝⁿ [subscript p], d ≠ 0. In the two dimensional case the argument is Lacey and McClain who made considerable refinements to this argument of Green who was bringing the argument to the finite field case from a paper of Shkredov. Let Ϝ ⁿ ₂ be the finite field of cardinality N = 2ⁿ. For all large N, any subset A⊂ Ϝⁿ ₂ × Ϝⁿ ₂ of cardinality ∣A ∣≳ N ² (log n) − [superscript epsilon], ε <, 1, must contain a corner {(x, y), (x + d, y), (x, y + d)} for x, y, d ∈ Ϝⁿ₂ and d ≠ 0.

Encontrar subestruturas aditivas que revelam um certo grau de organização em certos conjuntos contidos nos números naturais é o foco do estudo da combinatória aditiva. Desta área, resultados como os famosos Teorema de Van der Waerden e o Teorema de Szemerédi se destacam, revelando através de métodos combinatoriais que certas propriedades referentes ao tamanho de subconjuntos de inteiros implicam a existência de progressões aritméticas de tamanho arbitrariamente grande. Em meados de 1970, Furstenberg causou certa comoção no meio matemático ao publicar provas para ambos os teoremas usando métodos e ferramentas da teoria ergódica. Apesar de tal abordagem ter apresentado uma nova e profunda ligação entre as áreas, houve certa crítica pelo fato de não gerar resultados originais e por suas limitações (por exemplo, seus resultados costumam ser de caráter assintótico, sem lidar com limitantes e cotas, amplamente conhecidos pelos métodos combinatórios). Tais críticas foram silenciadas quando Ben Green e Terence Tao, usando tais métodos de teoria ergódica, demonstraram a incrível e bela afirmação de que os primos possuem progressões aritméticas de tamanho arbitrariamente grande, dando uma resposta definitiva para um enunciado conjecturado há muito tempo. Certamente, este foi um grande passo na matemática do século XXI. Deste então, novas abordagens foram amplamente estudadas e analisadas, de modo a aumentar ainda mais nossa compreensão sobre estes impressionantes conceitos. / Finding additive substructures that reveal a certain degree of organization in certain sets contained in the set of the natural numbers is the focus of the study of additive combinatorics. From this area, results such as the famous Van der Waerdens Theorem and Szemerédis Theorem stand out, revealing through combinatorial methods that certain properties concerning the size of subsets of integers imply the existence of arbitrarily long arithmetic progressions. In the mid-1970s Furstenberg caused some commotion in the mathematical world by publishing proofs for both theorems using methods and tools of ergodic theory rather than combinatorial methods. Although this approach had presented a new and deep link between those areas, there was some criticism for the lack of original results and some limitations of this technique (for instance, its results usually have an asymptotic flavour without dealing with bounds widely known by combinatorial methods). Such criticisms were silenced when Ben Green and Terence Tao, using such methods of ergodic theory, demonstrated the incredible and beautiful theorem that the primes have arithmetic progressions of arbitrarily large size, giving a definitive answer to a statement conjectured a long time ago. Certainly, this was a major step for the mathematics of the 21st century. Hence, new approaches have been extensively studied and analyzed in order to further increase our understanding of these impressive concepts.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-11-12T10:22:38Z No. of bitstreams: 2 Dissertação - Marcelo de Souza Melo - 2018.pdf: 4311410 bytes, checksum: b1e7a39d72be1f9d1a6d57405fb46c93 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-11-13T09:12:22Z (GMT) No. of bitstreams: 2 Dissertação - Marcelo de Souza Melo - 2018.pdf: 4311410 bytes, checksum: b1e7a39d72be1f9d1a6d57405fb46c93 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-11-13T09:12:22Z (GMT). No. of bitstreams: 2 Dissertação - Marcelo de Souza Melo - 2018.pdf: 4311410 bytes, checksum: b1e7a39d72be1f9d1a6d57405fb46c93 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-10-10 / This work shows how the content Arithmetic Progressions can be approached following the constructivist line of teaching, making the students have more active participation in the construction of their knowledge. It is veri_ed that using this model, one can improve students' understanding by introducing in the initial classes one or more problem situations in order to raise previous knowledge for the later acquisition of new knowledge. There are some arguments of professors / educators on this subject and also the practical application of classes structured in the constructivist line on arithmetic progressions, for students of the second year of high school in a public school in the Federal District. The observations about this style of class were made not only by the teacher who applied the activity proposed in class, but also by the students who answered questions that allowed to express the impressions about the activity. / Este trabalho mostra como o conteúdo Progressões Aritméticas pode ser abordado seguindo a linha construtivista de ensino, fazendo com que os alunos tenham participa ção mais ativa na construção do seu conhecimento. É veri_cado que utilizando esse modelo, pode-se melhorar a compreensão dos discentes, introduzindo nas aulas iniciais, uma ou mais situações-problema, com o intuito de levantar conhecimentos prévios para a aquisição posterior do novo saber. Existem algumas argumentações de professores/ educadores consagrados sobre esse tema e também a aplicação prática de aulas estruturadas na linha construtivista sobre progressões aritméticas, para alunos do segundo ano do ensino médio de uma escola pública do Distrito Federal. As observações sobre este estilo de aula foram feitas não somente pelo professor que aplicou a atividade proposta em sala aula, mas também pelos discentes que responderam questões que permitiam expressar as impressões sobre a atividade.

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Neste estudo, apresentamos conteúdos matemáticos adaptáveis tanto para os anos finais do ensino fundamental quanto para o ensino médio. Iniciamos com um conjunto de ideias preliminares: indução matemática, triângulo de Pascal, Binômio de Newton e relações trigonométricas, para a obtenção de fórmulas de somas finitas, em que os valores das parcelas são computados sobre números inteiros consecutivos, e da técnica de transformação de soma finita em telescópica. Enunciamos Progressões Aritméticas e Geométricas como sequências numéricas e suas propriedades, obtendo a soma de seus n primeiros termos, associando com propriedades do triângulo de Pascal. Por fim, descrevemos Funções Aritméticas, Funções Aritméticas Totalmente Multiplicativas e Fortemente Multiplicativas, como sequências de números naturais, com suas operações e propriedades, direcionando ao objetivo de calcular o número de divisores naturais de n, a soma de todos os divisores naturais de n, e assim por diante. Como consequência, exibimos a fórmula de contagem do número de polinômios mônicos irredutíveis. / In this study, we present mathematical content that is adaptable to both of the final years of elementary school and to high school. We start with a set of preliminary ideas: mathematical induction, Pascal\'s triangle, Newton\'s binomial and trigonometric relations, to obtain finite sum formulas, where the parts are computed on consecutive integers, and the technique for transforming a finite sum in telescopic one. We state the Arithmetic and Geometric Progressions as numerical sequences and study their properties, obtaining the sum of their n first terms, associating with properties of the Pascal\'s triangle. Finally, we describe the Arithmetic, Totally Multiplicative and Strongly Multiplicative Arithmetic Functions, as sequences of natural numbers, with their operations and properties, as a way to calculating the number of natural divisors of n, the sum of all natural divisors of n, and so on. As a consequence, we obtain the counting formula of the number of irreducible mononical polynomials.