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
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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.
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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.
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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
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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
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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