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Tel Aviv University
Raymond and Beverly Sackler
Faculty of Exact Sciences
School of Mathematical Sciences

Topics in Positional Games

THESIS SUBMITTED FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY by

Asaf Ferber

The research work for this thesis has been carried out

under the supervision of Prof. Michael Krivelevich

Submitted to the senate of Tel Aviv University

July 2013

iii

Acknowledgements

It would not have been possible to write this dissertation without the help and support of the kind people around me, to only some of whom it is possible to give particular mention

here.

Above all, I would like to thank the one and only, my supervisor, Professor Michael

Krivelevich, for his guidance, patience and continuous support throughout my studies. You

are my role model of a great mathematician and a great teacher. It was a great honor to be

your student.

Special thanks to Danny Hefetz, who took me under his wings, acted as a second supervisor for me, and also turned into a close friend. I am very grateful for all you have done for me and I hope to continue this legacy with the younger generation.

Moreover, I would like to thank all the professors from whom I had the pleasure and honor of learning. In particular, I would like to thank Noga Alon, Moti Gitik, Michael

Krivelevich, Benny Sudakov and Tibor Szab´o.

Next, I would like to thank all of my co-authors from all around the world. It was a great joy and honor to work with each of you! In particular I would like to thank my fellow

graduate students from Berlin, Dennis Clemens and Anita Liebenau, for the continuous inspiring collaboration.

I also wish to thank all my friends from the university for making my time there more enjoyable. In particular, I would like to thank Lev Buhovsky, Yaniv Dvir, Gal Kronenberg,

Alon Naor, Edva Roditty-Gershon, Wojciech Samotij and Amit Weinstein.

Of course, I would also like to thank my parents Malka and Arie, my brothers Benny and
Gadi, and my sister Daniela for their unequivocal support, for which my mere expression of

thanks does not suffice.

Last but not least, I would like to thank my lovely wife Shiran for her love and support.
I could not have asked for a better partner for life.

iv

Abstract

Positional Games is a rapidly evolving, relatively young topic in Combinatorics that is deeply linked to several popular areas of Mathematics and Theoretical Computer Science, such as Ramsey Theory, Classical Game Theory and Algorithms. Positional games are finite games

of complete information with no chance moves. Therefore, in theory, one can solve them completely by a finite (though absurdly large) case study. Note that this case study yields an optimal deterministic strategy for each such game. In practice, this is impossible as for example, given some initial board position in HEX, which is a positional game, it is PSPACE-complete to decide who wins. Clearly, from the view point of Combinatorics as well as Complexity Theory this is not the end, but rather the starting point of research. It turns out that, even though brute force is impractical, even for such naive looking games as 5 × 5 × 5 Tic-Tac-Toe, there exists in fact a very successful theory. This theory has experienced explosive growth in recent years, and for a systematic study of this subject the reader is referred to the excellent book by J. Beck [9].

The beginning of the systematic study of the theory of Positional games can be definitely attributed to two classical papers: that of Hales and Jewett from 1963 [36], and that of Erdo˝s and Selfridge from 1973 [24]. The significance of these two papers certainly cannot be

overestimated and goes far beyond the realm of Positional Games: the Hales-Jewett theorem is one of the cornerstones of modern Ramsey Theory, the Erd˝os-Selfridge argument was essentially the first derandomization procedure, a central concept in the theory of algorithms.

This thesis contains several novel contributions to the theory of Positional Games. One

major contribution is to the theory of Strong games (“who does it first?” games). We show a connection between fast strategies in weak games and winning strategies in strong games. We

then solve few natural strong games by finding explicit winning strategies. This is surprising

since these games are known to be notoriously hard to analyze and not much is known. Another major contribution is to the study of Positional Games played on random boards, initiated by Stojakovi´c and Szab´o in [59]. In this thesis we significantly improve the set of tools and enlarge our knowledge in the study of games played on the edge set of typical

random graphs. Such games are also interesting as purely random graph theoretic results

since they can be also viewed as a “measurement of robustness” of random models with

respect to some natural graph properties.

  • v
  • vi

Contents

  • 1 Introduction
  • 1

  • I Fast weak games VS strong games
  • 9

  • 2 The strong Hamiltonicity and perfect matching games
  • 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The Perfect Matching Game . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 The Hamilton Cycle Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

  • 3 The k-connectivity game
  • 23

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 A family of k-vertex-connected graphs . . . . . . . . . . . . . . . . . . . . . 25 3.3 Auxiliary games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 A large matching game . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.2 A weak positive minimum degree game . . . . . . . . . . . . . . . . . 29 3.3.3 A strong positive minimum degree game . . . . . . . . . . . . . . . . 31
3.4 The Maker-Breaker k-vertex-connectivity game . . . . . . . . . . . . . . . . 33 3.5 The strong k-vertex-connectivity game . . . . . . . . . . . . . . . . . . . . . 38 3.6 Concluding remarks and open problems . . . . . . . . . . . . . . . . . . . . . 41

  • 4 The specific tree game
  • 43

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

vii

4.1.1 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 Playing several biased games in parallel . . . . . . . . . . . . . . . . . 47 4.2.2 A perfect matching game . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2.3 A Hamiltonicity game . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Embedding a spanning tree quickly . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.1 Following Maker’s strategy for Case I . . . . . . . . . . . . . . . . . . 53 4.3.2 Following Maker’s strategy for Case II . . . . . . . . . . . . . . . . . 56
4.4 Concluding remarks and open problems . . . . . . . . . . . . . . . . . . . . . 59

  • 5 Very fast specific tree game
  • 61

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.1 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Trees which admit a long bare path . . . . . . . . . . . . . . . . . . . . . . . 64 5.3 Trees which do not admit a long bare path . . . . . . . . . . . . . . . . . . . 68 5.4 Building trees in optimal time . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.5 Concluding remarks and open problems . . . . . . . . . . . . . . . . . . . . . 86

  • II Games on random boards
  • 89

  • 6 Introduction
  • 91

6.0.1 Random graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

  • 7 Hitting time results
  • 95

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.1.1 Our results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.1.2 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . 98 7.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.2 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2.1 Basic positional games results . . . . . . . . . . . . . . . . . . . . . . 99 7.2.2 (R, c)-expanders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3 An expander game on pseudo-random graphs . . . . . . . . . . . . . . . . . . 101 7.4 Properties of random graphs and random graph processes . . . . . . . . . . . 105

viii

7.5 Hitting time of the k-vertex connectivity and perfect matching games . . . . 110
7.5.1 k-vertex connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.5.2 Perfect matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.6 Hitting time of the Hamiltonicity game . . . . . . . . . . . . . . . . . . . . . 113 7.7 Remarks on possible generalizations . . . . . . . . . . . . . . . . . . . . . . . 116

  • 8 Fast Maker-Breaker games on random boards
  • 117

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.1.1 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . 119
8.2 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.2.1 Basic positional games results . . . . . . . . . . . . . . . . . . . . . . 119 8.2.2 General graph theory results . . . . . . . . . . . . . . . . . . . . . . . 121 8.2.3 Properties of G(n, p) . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.2.4 Expanders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.3 The Perfect Matching Game . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.4 The Hamiltonicity Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8.5 The k-Connectivity Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8.6 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

  • 9 Biased games on random boards
  • 135

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    The Critical Bias for the Hamiltonicity Game Is Ln 푛

    (1+표(1))푛 The Critical Bias for the Hamiltonicity Game is ln 푛 Aadil Shaikh, Rohit Chakravorty and Thai Ling April 2020 Abstract: This paper serves as an exposition on the theorem proved by Michael Krivelevich which states that the critical bias for the Hamiltonicity game between a Maker and Breaker (1+표(1))푛 can be generalised as [1]. Initially, basic introductions into some concepts used in ln 푛 Combinatorial Game Theory and Graph Theory are given. These are then built upon to create lemmas and theorems. Using these tools, the result is proved in multiple stages. 1. Introduction Popular two-person strategy games such as Tic-Tac-Toe (Noughts and Crosses) and Hex can be generalised as having a position where the players take turns making moves to achieve a defined winning condition. This gives way to the notion of the positional game used in Combinatorial Game Theory (CGT) which can be described by the following conditions: • 푋 - the board, with a finite set of elements known as positions • ℱ - the winning-sets, which are a family of subsets of 푋 • A criterion for winning the game Definition 1. A maker-breaker game is a type of positional game where two players, the Maker and the Breaker take it in turns to take unclaimed elements on a board. The objective of the Maker is to hold all the elements of a winning set, whereas the Breaker wins if they can prevent this (i.e. hold at least one element in each winning set). Alternatively, the Maker-Breaker game can be more formally defined as: a triple (퐻, 푎, 푏), with 퐻 = (푋, ℱ) where 퐻 is a hypergraph.
  • Problems in Positional Games and Extremal Combinatorics

    Problems in Positional Games and Extremal Combinatorics

    Problems in Positional Games and Extremal Combinatorics Christopher Kusch Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften am Fachbereich f¨urMathematik und Informatik der Freien Universit¨atBerlin Berlin, 2017 ii Erstgutachter: Prof. Tibor Szab´o,PhD (Betreuer) Zweitgutachter: Prof. Dr. Anusch Taraz Tag der Disputation: 4.10.2017 Contents Preface ix Acknowledgement xi 1 Introduction1 1.1 Strong Ramsey games...............................2 1.1.1 Introduction to strong games.......................2 1.1.2 The strong Ramsey game.........................3 1.1.3 The transition to the infinite board....................3 1.2 Maker-Breaker G-games and van der Waerden games..............5 1.2.1 Introduction to Maker-Breaker games..................5 1.2.2 The Maker-Breaker G-game........................7 1.2.3 The van der Waerden game and its generalisation............9 1.2.4 General winning criteria.......................... 11 1.3 Hypergraphs with Property O ........................... 13 1.3.1 Introduction of the problem........................ 13 1.3.2 Results: an improved upper bound.................... 15 1.4 Shattering extremal families............................ 15 1.4.1 Shattering.................................. 15 1.4.2 The elimination conjecture........................ 18 1.4.3 The results................................. 18 2 Strong Ramsey games: Drawing on an infinite board 21 2.1 Overview of the proof............................... 21 2.2 Sufficient conditions for a draw.......................... 22 2.2.1 No threat.................................. 24 2.2.2 Special threat................................ 25 2.2.3 Standard threats.............................. 27 iii iv CONTENTS 2.3 An explicit construction.............................. 29 2.4 Concluding remarks and open problems..................... 37 3 General Winning Criteria for Maker-Breaker games 39 3.1 The threshold bias of the k-AP game......................
  • Positional Games Involve Two Players Alternately Claiming Unoccupied Elements of a Set X, the Board of the Game; the Elements of X Are Called Vertices

    Positional Games Involve Two Players Alternately Claiming Unoccupied Elements of a Set X, the Board of the Game; the Elements of X Are Called Vertices

    Positional Games Michael Krivelevich Abstract. Positional games are a branch of combinatorics, researching a variety of two- player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions. Mathematics Subject Classification (2010). Primary 05C57, 91A46; Secondary 05C80, 05D05, 05D10, 05D40. Keywords. Positional games, Ramsey theory, extremal set theory, probabilistic intu- ition. 1. Introductory words Positional games are a combinatorial discipline, whose basic aim is to provide a mathematical foundation for analysis of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. Though the field has been in existence for several decades, motivated partly by its recreational side, it advanced tremendously in the last few years, maturing into one of the central branches of modern combinatorics. It has been enjoying mutual and fruitful interconnections with other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic arXiv:1404.2731v1 [math.CO] 10 Apr 2014 combinatorics, as well as theoretical computer science. The aim of this survey is two-fold. It is meant to provide a brief, yet gentle, introduction to the subject to those with genuine interest and basic knowledge in combinatorics. At the same time, we cover recent progress in the field, as well as its standing challenges and open problems.
  • On the Threshold for the Maker-Breaker H-Game

    On the Threshold for the Maker-Breaker H-Game

    ON THE THRESHOLD FOR THE MAKER-BREAKER H-GAME RAJKO NENADOV1, ANGELIKA STEGER1 AND MILOSˇ STOJAKOVIC´ 2 Abstract. We study the Maker-Breaker H-game played on the edge set of the random graph Gn;p. In this game two players, Maker and Breaker, alternately claim unclaimed edges of Gn;p, until all edges are claimed. Maker wins if he claims all edges of a copy of a fixed graph H; Breaker wins otherwise. In this paper we show that, with the exception of trees and triangles, the threshold for an H-game is given by the threshold of the corresponding Ramsey property of Gn;p with respect to the graph H. Keywords. Positional games; random graphs; Maker-Breaker 1. Introduction Combinatorial games are games like Tic-Tac-Toe or Chess in which each player has perfect information and players move sequentially. Outcomes of such games can thus, at least in principle, be predicted by enumerating all possible ways in which the game may evolve. But, of course, such complete enumerations usually exceed available computing powers, which keeps these games interesting to study. In this paper we take a look at a special class of combinatorial games, the so- called Maker-Breaker positional games. Given a finite set X and a family E of subsets of X, two players, Maker and Breaker, alternate in claiming unclaimed elements of X until all the elements are claimed. Unless explicitly stated otherwise, Maker starts the game. Maker wins if he claims all elements of a set from E, and Breaker wins otherwise. The set X is referred to as the board, and the elements of E as the winning sets.
  • Positional Games

    Positional Games

    Mathematisches Forschungsinstitut Oberwolfach Report No. 44/2018 DOI: 10.4171/OWR/2018/44 Mini-Workshop: Positional Games Organized by Dan Hefetz, Ariel Michael Krivelevich, Tel Aviv Milos Stojakovic, Novi Sad Tibor Szabo, Berlin 30 September – 6 October 2018 Abstract. This mini-workshop focused on Positional Games and related fields. Positional Games Theory is a branch of Combinatorics whose main aim is to systematically develop an extensive mathematical basis for a variety of two-player games of perfect information and without chance moves, usually played on discrete objects. These include popular recreational games such as Tic-Tac-Toe and Hex as well as purely abstract games played on graphs and hypergraphs. Though a close relative of the classical Game Theory of von Neumann and of Nim-like games, popularized by Conway and others, Po- sitional Games are quite different and are more of a combinatorial nature. The subject is strongly related to several other branches of Combinatorics like Ramsey Theory, Extremal Graph and Set Theory, and the Probabilistic Method. It has also proven to be instrumental in deriving central results in Theoretical Computer Science, in particular in derandomization and algorith- mization of important probabilistic tools. Despite being a relatively young topic, there are already three textbooks dedicated to Positional Games as well as one invited talk at the International Congress of Mathematicians. Dur- ing this mini-workshop, several new exciting developments in the field were presented and discussed. We have also made some progress towards solving various open problems in Positional Games Theory and related areas. Mathematics Subject Classification (2010): 91A24 (Positional games), 05C57 (Games on graphs), 91A43 (Games involving graphs), 91A46 ()Combinatorial games), 05C65 (Hypergraphs), 05C80 (Random graphs), 05D40 (Probabilistic methods), 05D10 (Ramsey theory).
  • New Methods and Boards for Playing Tic-Tac-Toe

    New Methods and Boards for Playing Tic-Tac-Toe

    University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 2012 Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe Mary Jennifer Riegel The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Riegel, Mary Jennifer, "Nontraditional Positional Games: New methods and boards for playing Tic-Tac- Toe" (2012). Graduate Student Theses, Dissertations, & Professional Papers. 707. https://scholarworks.umt.edu/etd/707 This Dissertation is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe By Mary Jennifer Riegel B.A., Whitman College, Walla Walla, WA, 2006 M.A., The University of Montana, Missoula, MT, 2008 Dissertation presented in partial fulfillment of the requirements for the degree of Doctorate of Philosophy in Mathematics The University of Montana Missoula, MT May 2012 Approved by: Sandy Ross, Associate Dean of the Graduate School Graduate School Dr. Jennifer McNulty, Chair Mathematical Sciences Dr. Mark Kayll Mathematical Sciences Dr. George McRae Mathematical Sciences Dr. Nikolaus Vonessen Mathematical Sciences Dr. Michael Rosulek Computer Science Riegel, Mary J., Doctorate of Philosophy, May 2012 Mathematics Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe Committee Chair: Jennifer McNulty, Ph.D.
  • A Threshold for the Maker-Breaker Clique Game∗

    A Threshold for the Maker-Breaker Clique Game∗

    A threshold for the Maker-Breaker clique game∗ Tobias M¨uller y MiloˇsStojakovi´c z Abstract We study the Maker-Breaker k-clique game played on the edge set of the random graph G(n; p). In this game, two players, Maker and Breaker, alternately claim un- claimed edges of G(n; p), until all the edges are claimed. Maker wins if he claims all the edges of a k-clique; Breaker wins otherwise. We determine that the threshold for the − 2 graph property that Maker can win this game is at n k+1 , for all k > 3, thus proving a conjecture from [10]. More precisely, we conclude that there exist constants c; C > 0 − 2 − 2 such that when p > Cn k+1 the game is Maker's win a.a.s., and when p < cn k+1 it is Breaker's win a.a.s. For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker's win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K5 with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker's win in the triangle game played on the edge set of G(n; p). 1 Introduction Let X be a finite set and let F ⊆ 2X be a family of subsets of X. In the positional game (X; F), two players take turns in claiming one previously unclaimed element of X.