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 super- visor 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 Erd}osand 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-Selfridgeargument 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´cand Szab´oin [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 9.1 Introduction . 135 9.1.1 Notation and terminology . 137 9.2 Auxiliary results . 138 9.2.1 Binomial distribution bounds . 138 9.2.2 Basic positional games results . 138 9.2.3 (R; c)-Expanders . 140 9.2.4 Properties of G ∼ G(n; p)........................ 141 9.2.5 The minimum degree game . 148 9.3 Maker-Breaker games on G(n; p)......................... 152 9.3.1 Breaker's win . 152 9.3.2 Maker's win . 153 9.4 Avoider-Enforcer games on G(n; p)....................... 156 9.4.1 Avoider's win . 156 ix 9.4.2 Enforcer's win . 157 9.5 Concluding remarks and open questions . 159 III The odd cycle game 161 10 The odd cycle game 163 10.1 Introduction . 163 10.1.1 Notation and terminology . 165 10.2 Auxiliary results . 165 10.3 Proofs of Theorems 10.3 and 10.4 . 168 10.4 Proof of Theorem 10.5 . 169 x Chapter 1 Introduction Positional games are two player games played on any hypergraph F ⊆ 2X . The set X is referred to as the board of the game and the subsets of F are the target sets. The players, I and II, alternately claim a and b previously unclaimed elements of the board X, respectively. The game ends when every element of X has been claimed by the players. For every a; b and F, we denote this game by (a; b; F) or call it the (a : b) game F, or the (a : b) game (X; F) (in the most basic version a = b = 1 { the so called unbiased game, we denote the game as the game F or the game (X; F)). The game is specified completely by defining who wins in every possible game scenario.