DOCSLIB.ORG

Introduction to Ramsey Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Introduction to Ramsey Theory Lecture notes for undergraduate course Introduction to Ramsey Theory Lecture notes for undergraduate course Veselin Jungic Simon Fraser University Production Editor: Veselin Jungic Edition: Second Edition ©2014-–2021 Veselin Jungic This work is licensed under the Creative Commons Attribution-NonCommercialShareAlike 4.0 International License. Youcan view a copy of the license at http://creativecommons.org/ licenses/by-nc-sa/4.0/. To my sons, my best teachers. - Veselin Jungic Acknowledgements Drawings of Alfred Hale, Robert Jewett, Richard Rado, Frank Ramsey, Issai Schur, and Bartel van der Waerden were created by Ms. Kyra Pukanich under my directions. The drawing of Frank Ramsey displayed at the end of Section 1.3 was created by Mr. Simon Roy under my directions. I am grateful to Professor Tom Brown, Mr. Rashid Barket, Ms. Jana Caine, Mr. Kevin Halasz, Ms. Arpit Kaur, Ms. Ha Thu Nguyen, Mr. Andrew Poelstra, Mr. Michael Stephen Paulgaard, and Ms. Ompreet Kaur Sarang for their comments and suggestions which have been very helpful in improving the manuscript. I would like to acknowledge the following colleagues for their help in creating the PreTeXt version of the text. • Jana Caine, Simon Fraser University • David Farmer, American Institute of Mathematics • Sean Fitzpatrick , University of Lethbridge • Damir Jungic, Burnaby, BC I am particularly thankful to Dr. Sean Fitzpatrick for inspiring and encouraging me to use PreTeXt. Veselin Jungic vii Preface The purpose of these lecture notes is to serve as a gentle introduction to Ramsey theory for those undergraduate students interested in becoming familiar with this dynamic segment of contemporary mathematics that combines, among others, ideas from number theory and combinatorics. Since this booklet contains the class lecture notes, the reader will occasionally need the help of a more knowledgeable other: an instructor, a peer, a book, or Google. In addition to the bibliography, links with the relevant freely available online resources are provided at the end of each section. The only real prerequisites to fully grasp the material presented in these lecture notes, to paraphrase Professor Fikret Vajzović (1928 — 2017), is knowing how to read and write and possessing a certain level of mathematical maturity. Any undergraduate student who has successfully completed the standard calculus sequence of courses and a standard first (or second) year linear algebra course and has a genuine interest in learning mathematics should be able to master the main ideas presented here. My wish is to give to the reader both challenging and enjoyable experiences in learning some of the basic facts about Ramsey theory, a relatively new mathematical field. But what is Ramsey theory? Probably the best-known description of Ramsey theory is provided by Theodore S. Motzkin: Complete disorder is impossible. Here are a few more: • Ramsey theory studies the mathematics of colouring. — Alexander Soifer • Ramsey theory is the study of the preservation of properties under set partitions. — Bruce Landman and Aaron Robertson • The fundamental kind of question Ramsey theory asks is: can one always find order in chaos? If so, how much? Just how large a slice of chaos do we need to be sure to find a particular amount of order in it? — Imre Leader • If mathematics is a science of patterns, then Ramsey theory is a science of the stubbornness of patterns. – V. Jungic No project such as this can be free from errors and incompleteness. I would be grateful to anyone who points out any typos, errors, or provides any other suggestion on how to improve this manuscript. Veselin Jungic Department of Mathematics, Simon Fraser University Contact address: [email protected] In Burnaby, B.C., November 2020 viii Contents Acknowledgements vii Preface viii 1 Introduction: Pioneers and Trailblazers 1 1.1 Complete Chaos is Impossible..............1 1.2 Paul Erdős.....................4 1.3 Frank Plumpton Ramsey................9 2 Ramsey’s Theorem 13 2.1 The Pigeonhole Principle................ 13 2.2 Ramsey’s Theorem: Friends and Strangers.......... 16 2.3 Ramsey’s Theorem: Two Colours............. 20 2.4 Ramsey’s Theorem, Infinite Case............. 26 2.5 Exercises..................... 29 3 van der Waerden’s Theorem 37 3.1 Bartel van der Waerden................ 37 3.2 van der Waerden’s Theorem: 3–term APs.......... 41 3.3 Proof of van der Waerden’s Theorem............ 46 3.4 van der Waerden’s Theorem: How Far and Where?....... 51 3.5 van der Waerden’s Theorem: A Few Related Questions...... 56 3.6 Exercises..................... 59 4 Schur’s Theorem and Rado’s Theorem 71 4.1 Issai Schur.................... 71 4.2 Schur’s Theorem.................. 74 4.3 Richard Rado.................... 79 4.4 Rado’s Theorem................... 82 4.5 Exercises..................... 87 5 The Hales-Jewett Theorem 95 5.1 Combinatorial Lines................. 95 5.2 The Hales-Jewett Theorem............... 98 ix CONTENTS x 5.3 Exercises.....................106 6 Colourings of the Plane 115 6.1 Erdős-Szekeres Problem of Convex Polygons.........115 6.2 Erdős-Szekeres Problem of Convex Polygons - Part Two.....120 6.3 The Chromatic Number of the Plane............127 6.4 The Polychromatic Number of the Plane...........137 6.5 Fractional Chromatic Number..............142 6.6 Exercises.....................144 Bibliography 156 Chapter 1 Introduction: Pioneers and Trail- blazers 1.1 Complete Chaos is Impossible Complete disorder is impossible. — Theodore S. Motzkin, Israeli- American mathematician, 1908 — 1970. What is Ramsey theory? • Ramsey theory is the mathematics of colouring. - Soifer • Ramsey theory is the study of the preservation of properties under set partitions. - Landman-Robertson • The fundamental kind of question Ramsey theory asks is: can one always find order in chaos? If so, how much? Just how large a slice of chaos do we need to be sure to find a particular amount of order in it? - Leader • If mathematics is a science of patterns, then Ramsey theory is a science of the stubbornness of patterns. - Jungic Example 1.1.1 A Ramsey theory problem: If the natural numbers are finitely coloured, i.e. the set of natural numbers is partitioned into a finite number of cells, must there exist x, y (with x and y not both equal to 2) with x + y and xy monochromatic, i.e., x + y and xy belong to the same partition cell? (See Figure 1.1.2.) x + y x · y x ? ) y N N Figure 1.1.2 Monochromatic pattern? The problem was posed by Neil Hindman in the late 1970s and resolved by Joel Moreira in 2017: Monochromatic sums and products in N, Annals of Mathematics (2) 185 (2017), no. 3, 1069–1090. [ arXiv] What makes this problem to be a typical Ramsey theory problem is the following: 1 CHAPTER 1. INTRODUCTION: PIONEERS AND TRAILBLAZERS 2 • The topic: the problem is to determine the relationship between the set of all finite partitions of the natural numbers and a certain pattern. • The fact that any numerically literate person can understand the problem. • It is a difficult problem. Example 1.1.3 Schur’s Theorem: For any partition of the positive integers into a finite number of parts, one of the parts contains three integers x; y; z with x + y = z. (See Figure 1.1.5.) Figure 1.1.4 Issai Schur (1875 — 1941) ) x y x + y N N Figure 1.1.5 True, by Schur’s theorem Example 1.1.6 van der Waerden’s Theorem - Special Case. For any partition of the positive integers into a finite number of parts, one of the parts contains three integers x; y; z with x + y = 2z. (See Figure 1.1.8.) Figure 1.1.7 Bar- tel Leendert van der Waerden (1903 — 1996) CHAPTER 1. INTRODUCTION: PIONEERS AND TRAILBLAZERS 3 ) x+y x 2 y N N Figure 1.1.8 True, by van der Waerden’s theorem Example 1.1.9 Rado’s Theorem - Special Case. For any partition of the positive integers into a finite number of parts, one of the parts contains three integers x; y; z with ax + by + cz = 0, a , 0, b , 0, c , 0, if and only if one of the following conditions holds a + b + c = 0 or a + b = 0 or a + c = 0 or b + c = 0. (See Figure 1.1.11.) Figure 1.1.10 Richard Rado (1906 — 1989) 2 a + b + c = 0 or 3 6 7 6 a + b = 0 or 7 + 6 7 ) y 6 a + c = 0 or 7 x − ax+by 6 7 c 6 b + c = 0 7 4 5 N N Figure 1.1.11 True, by Rado’s theorem Example 1.1.12 Ramsey’s Theorem - Special case. If there are at least six people at dinner then either there are three mutual acquaintances or there are three mutual strangers. (See Figure 1.1.14.) Figure 1.1.13 Frank Plumpton Ramsey (1903 — 1930) CHAPTER 1. INTRODUCTION: PIONEERS AND TRAILBLAZERS 4 What’s next? Blue = acquaintances; Red = strangers. Figure 1.1.14 Proof of a special case of Ramsey’s theorem. Example 1.1.15 Hales-Jewett Theorem - Informal. In large enough di- mensions, the game of Tic-Tac-Toe cannot end in a draw. Figure 1.1.16 Alfred Hales and Robert Jewett × × 11 12 13 × ) × , 21 22 23 × × × × 31 32 33 Tic-Tac-Toe It’s a draw! Same but different Figure 1.1.17 Tic-Tac-Toe Tic-Tac-Toe - It is a win! 11 12 13 11 12 13 11 12 13 21 22 23 or 21 22 23 or 21 22 23 31 32 33 31 32 33 31 32 33 Resources. 1. See [2], [3], and [7]. 2. Ramsey Theory - Wikipedia 3. Ramsey Theory by R. Graham and B. Rothschild 4. Ramsey Theory by J.
Recommended publications
  • LINEAR ALGEBRA METHODS in COMBINATORICS László Babai

    LINEAR ALGEBRA METHODS in COMBINATORICS László Babai

    LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students.
  • Session 9: Pigeon Hole Principle and Ramsey Theory - Handout

    Session 9: Pigeon Hole Principle and Ramsey Theory - Handout

    Session 9: Pigeon Hole Principle and Ramsey Theory - Handout “Suppose aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack.” image: Frank P. Ramsey (1903-1926) quote: Paul Erd¨os (1913-1996) The following dots are in general position (i.e. no three lie on a straight line). 1) If it exists, find a convex quadrilateral in this cluster of dots. 2) If it exists, find a convex pentagon in this cluster of dots. 3) If it exists, find a convex hexagon in this cluster of dots. 4) If it exists, find a convex heptagon in this cluster of dots. 5) If it exists, find a convex octagon in this cluster of dots. 6) Draw 8 points in general position in such a way that the points do not contain a convex pentagon. 7) In my family there are 2 adults and 3 children. When our family arrives home how many of us must enter the house in order to ensure there is at least one adult in the house? 8) Show that among any collection of 7 natural numbers there must be two whose sum or difference is divisible by 10. 9) Suppose a party has six people.
  • Pioneers of Representation Theory, by Charles W

    Pioneers of Representation Theory, by Charles W

    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 37, Number 3, Pages 359{362 S 0273-0979(00)00867-3 Article electronically published on February 16, 2000 Pioneers of representation theory, by Charles W. Curtis, Amer. Math. Soc., Prov- idence, RI, 1999, xvi + 287 pp., $49.00, ISBN 0-8218-9002-6 The theory of linear representations of finite groups emerged in a series of papers by Frobenius appearing in 1896{97. This was at first couched in the language of characters but soon evolved into the formulation now considered standard, in which characters give the traces of representing linear transformations. There were of course antecedents in the number-theoretic work of Lagrange, Gauss, and others| especially Dedekind, whose correspondence with Frobenius suggested a way to move from characters of abelian groups to characters of arbitrary finite groups. In the past century this theory has developed in many interesting directions. Besides being a natural tool in the study of the structure of finite groups, it has turned up in many branches of mathematics and has found extensive applications in chemistry and physics. Marking the end of the first century of the subject, the book under review offers a somewhat unusual blend of history, biography, and mathematical exposition. Before discussing the book itself, it may be worthwhile to pose a general question: Does one need to know anything about the history of mathematics (or the lives of individual mathematicians) in order to appreciate the subject matter? Most of us are complacent about quoting the usual sloppy misattributions of famous theorems, even if we are finicky about the details of proofs.
  • The Unit Distance Graph Problem and Equivalence Relations

    The Unit Distance Graph Problem and Equivalence Relations

    Math/CCS 103 Professor: Padraic Bartlett Lecture 11: The Unit Distance Graph Week 8 UCSB 2014 (Source: \The Mathematical Coloring Book," by Alexander Soifer.) 1 The Unit Distance Graph Problem 2 Definition. Consider the following method for turning R into a graph: 2 • Vertices: all points in R . • Edges: connect any two points (a; b) and (c; d) iff the distance between them is exactly 1. This graph is called the unit distance graph. Visualizing this is kinda tricky | it's got an absolutely insane number of vertices and edges. However, we can ask a question about it: Question. How many colors do we need in order to create a proper coloring of the unit distance graph? So: the answer isn't immediately obvious (right?) Instead, what we're going to try to do is just bound the possible answers, to get an idea of what the answers might be. How can we even bound such a thing? Well: to get a lower bound, it suffices to consider finite graphs G that we can draw in the plane using only straight edges of length 1. Because 2 our graph on R must contain any such graph \inside" of itself, examining these graphs will give us some easy lower bounds! So, by examining a equilateral triangle T , which has χ(T ) = 3, we can see that 2 χ(R ) ≥ 3: This is because it takes three colors to color an equilateral triangle's vertices in such a way that no edge has two endpoints of the same color. Similarly, by examining the following pentagonal construction (called a Moser spindle,) 1 we can actually do one better and say that 2 χ(R ) ≥ 4: Verify for yourself that you can't color this graph with three colors! 2 Conversely: to exhibit an upper bound on χ(R ) of k, it suffices to create a way of \painting" the plane with k-colors in such a way that no two points distance 1 apart get the same color.
  • Breakthrough in My Favorite Open Problem of Mathematics: Chromatic Number of the Plane Alexander Soifer

    Breakthrough in My Favorite Open Problem of Mathematics: Chromatic Number of the Plane Alexander Soifer

    Breakthrough in My Favorite Open Problem of Mathematics: Chromatic Number of the Plane Alexander Soifer [I] can’t offer money for nice problems of other people because then I will really go broke… It is a very nice problem. If it were mine, I would offer $250 for it. – Paul Erdős Boca Raton, February 1992 1. Chromatic Number of the Plane: The Problem On behalf of all enthusiasts of this problem, let me express gratitude to my friend, the late Edward Nelson, who created this problem at a tender age of 18 in November 1950: What is the smallest number of colors sufficient for coloring the plane in such a way that no two points of the same color are unit distance apart? This number is called the chromatic number of the plane and is denoted by . In 1961, the Swiss geometer Hugo Hadwiger admitted that he was not the author of the problem, even though the name “Harwiger-Nelson” got stuck to the problem, just as Cardano did not author the Cardano Formula, and Pythagoras Theorem was known a millennium before the great Greek was born. Such is life with credits in mathematics. Right at problem’s birth, Eddie Nelson determined the lower bound of 4, and his 20-year old friend John Isbell, 20 figured out the upper bound of 7: = 4, or 5, or 6, or 7 A very broad spread. Which one is the value of ? Paul Erd"os thought 5. On May 28, 2009, during the DIMACS Ramsey Theory International Workshop that I organized on request of DIMACS Director Fred Roberts, I asked the distinguished audience to determine the chromatic number of the plane by democratic means of a vote.
  • The Parameterized Complexity of Finding Point Sets with Hereditary Properties

    The Parameterized Complexity of Finding Point Sets with Hereditary Properties

    The Parameterized Complexity of Finding Point Sets with Hereditary Properties David Eppstein1 Computer Science Department, University of California, Irvine, USA [email protected] Daniel Lokshtanov2 Department of Informatics, University of Bergen, Norway [email protected] Abstract We consider problems where the input is a set of points in the plane and an integer k, and the task is to find a subset S of the input points of size k such that S satisfies some property. We focus on properties that depend only on the order type of the points and are monotone under point removals. We show that not all such problems are fixed-parameter tractable parameterized by k, by exhibiting a property defined by three forbidden patterns for which finding a k-point subset with the property is W[1]-complete and (assuming the exponential time hypothesis) cannot be solved in time no(k/ log k). However, we show that problems of this type are fixed-parameter tractable for all properties that include all collinear point sets, properties that exclude at least one convex polygon, and properties defined by a single forbidden pattern. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms Keywords and phrases parameterized complexity, fixed-parameter tractability, point set pattern matching, largest pattern-avoiding subset, order type 1 Introduction In this work, we study the parameterized complexity of finding subsets of planar point sets having hereditary properties, by analogy to past work on hereditary properties of graphs. In graph theory, a hereditary properties of graphs is a property closed under induced subgraphs.
  • Mathematicians Fleeing from Nazi Germany

    Mathematicians Fleeing from Nazi Germany

    Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D.
  • Some Remarks on Even-Hole-Free Graphs

    Some Remarks on Even-Hole-Free Graphs

    Some remarks on even-hole-free graphs Zi-Xia Song∗ Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Abstract A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger’s conjecture and the Erd˝os-Lov´asz Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all k ≥ 7, every even-hole-free graph with no Kk minor is (2k − 5)-colorable; every even-hole-free graph G with ω(G) < χ(G) = s + t − 1 satisfies the Erd˝os-Lov´asz Tihany conjecture provided that t ≥ s>χ(G)/3. Furthermore, we prove that every 9-chromatic graph G with ω(G) ≤ 8 has a K4 ∪ K6 minor. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs. 1 Introduction All graphs in this paper are finite and simple. For a graph G, we use V (G) to denote the vertex set, E(G) the edge set, |G| the number of vertices, e(G) the number of edges, δ(G) the minimum degree, ∆(G) the maximum degree, α(G) the independence number, ω(G) the clique number and χ(G) the chromatic number. A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges.
  • Math 126 Lecture 4. Basic Facts in Representation Theory

    Math 126 Lecture 4. Basic Facts in Representation Theory

    Math 126 Lecture 4. Basic facts in representation theory. Notice. Definition of a representation of a group. The theory of group representations is the creation of Frobenius: Georg Frobenius lived from 1849 to 1917 Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups, the representation theory of groups and the character theory of groups. Find out more at: http://www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Frobenius.html Matrix form of a representation. Equivalence of two representations. Invariant subspaces. Irreducible representations. One dimensional representations. Representations of cyclic groups. Direct sums. Tensor product. Unitary representations. Averaging over the group. Maschke’s theorem. Heinrich Maschke 1853 - 1908 Schur’s lemma. Issai Schur Biography of Schur. Issai Schur Born: 10 Jan 1875 in Mogilyov, Mogilyov province, Russian Empire (now Belarus) Died: 10 Jan 1941 in Tel Aviv, Palestine (now Israel) Although Issai Schur was born in Mogilyov on the Dnieper, he spoke German without a trace of an accent, and nobody even guessed that it was not his first language. He went to Latvia at the age of 13 and there he attended the Gymnasium in Libau, now called Liepaja. In 1894 Schur entered the University of Berlin to read mathematics and physics. Frobenius was one of his teachers and he was to greatly influence Schur and later to direct his doctoral studies. Frobenius and Burnside had been the two main founders of the theory of representations of groups as groups of matrices. This theory proved a very powerful tool in the study of groups and Schur was to learn the foundations of this subject from Frobenius.
  • SAT Approach for Decomposition Methods

    SAT Approach for Decomposition Methods

    SAT Approach for Decomposition Methods DISSERTATION submitted in partial fulfillment of the requirements for the degree of Doktorin der Technischen Wissenschaften by M.Sc. Neha Lodha Registration Number 01428755 to the Faculty of Informatics at the TU Wien Advisor: Prof. Stefan Szeider Second advisor: Prof. Armin Biere The dissertation has been reviewed by: Daniel Le Berre Marijn J. H. Heule Vienna, 31st October, 2018 Neha Lodha Technische Universität Wien A-1040 Wien Karlsplatz 13 Tel. +43-1-58801-0 www.tuwien.ac.at Erklärung zur Verfassung der Arbeit M.Sc. Neha Lodha Favoritenstrasse 9, 1040 Wien Hiermit erkläre ich, dass ich diese Arbeit selbständig verfasst habe, dass ich die verwen- deten Quellen und Hilfsmittel vollständig angegeben habe und dass ich die Stellen der Arbeit – einschließlich Tabellen, Karten und Abbildungen –, die anderen Werken oder dem Internet im Wortlaut oder dem Sinn nach entnommen sind, auf jeden Fall unter Angabe der Quelle als Entlehnung kenntlich gemacht habe. Wien, 31. Oktober 2018 Neha Lodha iii Acknowledgements First and foremost, I would like to thank my advisor Prof. Stefan Szeider for guiding me through my journey as a Ph.D. student with his patience, constant motivation, and immense knowledge. His guidance helped me during the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor. Next, I would like to thank my co-advisor Prof. Armin Biere, my reviewers, Prof. Daniel Le Berre and Dr. Marijn Heule, and my Ph.D. committee, Prof. Reinhard Pichler and Prof. Florian Zuleger, for their insightful comments, encouragement, and patience.
  • Single Digits

    Single Digits

    ...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library
  • Mechanical Proving for ERDÖS-SZEKERES Problem

    Mechanical Proving for ERDÖS-SZEKERES Problem

    2016 6th International Conference on Applied Science, Engineering and Technology (ICASET 2016) Mechanical Proving for ERDÖS-SZEKERES Problem 1Meijing Shan Institute of Information science and Technology, East China University of Political Science and Law, Shanghai, China. 201620 [email protected] Keywords: Erdös-Szekeres problem, Automated deduction, Mechanical proving Abstract：The Erdös-Szekeres problem was an open unsolved problem in computational geometry and related fields from 1935. Many results about it have been shown. The main concern of this paper is not only show how to prove this problem with automated deduction methods and tools but also contribute to the significance of automated theorem proving in mathematics using advanced computing technology. The present case is engaged in contributing to prove or disprove this conjecture and then solve this problem. The key advantage of our method is to utilize the mechanical proving instead of the traditional proof and this method could improve the arithmetic efficiency. Introduction The following famous problem has attracted more and more attention of many mathematicians [3, 6, 12, 16] due to its beauty and elementary character. Finding the exact value of N(n) turns out to be a very challenging problem. The problem is very easy to explain and understand. The Erdös-Szekeres Problem 1.1 [4, 15]. For any integer n ≥ 3, determine the smallest positive integer N(n) such that any set of at least N(n) points in generalposition in the plane contains n points that are the vertices of a convex n-gon. A set of points in the plane is said to be in the general position if it contains no three points on a line.