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研究生： 譚芮妲
Regina Ayunita Tarigan
論文名稱： Szemerédi’s Regularity Lemma and Its Applications
Szemerédi’s Regularity Lemma and Its Applications
指導教授： 沈俊嚴
Chun-Yen Shen
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2017
畢業學年度： 105
語文別： 英文
論文頁數： 40
中文關鍵詞： extremal graph 、partition graph 、arithmetic progression 、triangle removal lemma 、graph density 、random graphs 、ϵ-regularity 、ϵ-regular partition 、equipartition 、Szemerédi 、Regularity Lemma 、Roth's Theorem
外文關鍵詞： extremal graph, partition graph, arithmetic progression, triangle removal lemma, graph density, random graphs, ϵ-regularity, ϵ-regular partition, equipartition, Szemerédi, Regularity Lemma, Roth's Theorem
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    • 在本篇論文中, 我們研讀探討在圖論領域裡的著名定理 Szemerédi’s Regularity Lemma 以及此定理的應用. 簡單來說 Szemerédi’s Regularity Lemma 可以將一個圖分解成許多幾乎相等的分割, 而這些分割之間彼此兩兩幾乎是隨機的分佈. 最後我們將討論如何使用此定理運用在數論的一個著名的定理, 所謂的 Roth's Theorem. 此定理敘述任一個整數的子集合存在長度為三的等差數列只要此集合的密度大於零.

    In this dissertation, the Szemerédi’s Regularity Lemma and its application are studied. This lemma is used to partition a large enough graph into almost equal parts so that the number of edges across the parts is fairly random. On the other hand, Roth's Theorem states that there exists an arithmetic progression with length 3 in a subset in integer with positive upper density. We shall see that it can be proved by using triangle removal lemma, which is an application of Szemerédi’s Regularity Lemma.

    摘要 i Abstract ii Acknowledgementiii Contentsiv List of Figuresv 1 Introduction 1 1.1 A Brief Overview of GraphTheory............1 1.2 Some Necessary Terms...................3 1.3 The Degree of a Vertex...................5 1.4 Bipartite Graph.......................6 1.5 Clique and Independent Set................7 2 Extremal GraphTheory 8 2.1 Forbidden Subgraph Problem...............8 2.2 Complete Subgraph.....................8 2.3 Turán Graph (Tr(n)) ....................8 2.4 Erdös-Stone Theorem....................10 3 Szemerédi’s Regularity Lemma 11 3.1 Brief Overview.......................11 3.2 Szemerédi's Regularity Lemma...............13 4 Further Concept and Application of Szemerédi's Regu- larity Lemma 21 4.1 Some Historical Background................21 4.2 Extremal Problem.....................22 4.3 Removal Lemma......................24 4.4 Roth's Theorem in Arithmetic Progression (k = 3)....28 References 30

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