## An Irregular Mind: Szemerédi is 70Imre Bárány, Jozsef Solymosi Szemerédi's influence on today's mathematics, especially in combinatorics, additive number theory, and theoretical computer science, is enormous. This volume is a celebration of Szemerédi's achievements and personality, on the occasion of his seventieth birthday. It exemplifies his extraordinary vision and unique way of thinking. A number of colleagues and friends, all top authorities in their fields, have contributed their latest research papers to this volume. The topics include extension and applications of the regularity lemma, the existence of k-term arithmetic progressions in various subsets of the integers, extremal problems in hypergraphs theory, and random graphs, all of them beautiful, Szemerédi type mathematics. It also contains published accounts of the first two, very original and highly successful Polymath projects, one led by Tim Gowers and the other by Terry Tao. |

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### Contents

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PERCOLATION ON SELFDuAL POLYGON CONFIGURATIONS | 130 |

ON EXPONENTIAL SUMS IN FINITE FIELDS | 219 |

AN ESTIMATE OF INCOMPLETE MIXED CHARACTER SUMS | 243 |

CROSSINGS BETWEEN CURVES WITH MANY TANGENCIES | 251 |

REGULARITY PARTITIONS AND THE TOPOLOGY OF GRAPHONS | 415 |

EXTREMAL PROBLEMS FOR SPARSE GRAPHS | 447 |

SQUARES IN SUMSETS | 491 |

ARE THERE ARBITRARILY LONG ARITHMETIC PROGRESSIONS IN THE SEQUENCE OF TWIN PRIMES? | 525 |

DIRACTYPE QUESTIONS FOR HYPERGRAPHS A SURVEY OR MORE PROBLEMS FOR ENDRE TO SOLVE | 561 |

TOWARDS A NONCOMMUTATIVE PLUNNECKETYPE INEQUALITY | 591 |

QUASIRANDOM MULTITYPE GRAPHS | 606 |

PSEUDORANDOMNESS IN COMPUTER S CIENCE AND IN ADDITIVE COMBINATORICS | 619 |

AN ARITHMETIC REGULARITY LEMMA AN ASSOCIATED COUNTING LEMMA AND ApPLICATIONS | 261 |

YET ANOTHER PROOF OF SZEMEREDIS THEOREM | 335 |

ONLINE LINEAR DISCREPANCY OF PARTIALLY ORDERED SETS | 343 |

ON THE TRIANGLE R EMOVAL L EMMA FOR S UBGRAPHS OF SPARSE P SEUDORANDOM GRAPHS | 358 |

ALMOST ALL HFREE GRAPHS HAVE THE ERDOSHAJNAL PROPERTY | 405 |

INTRODUCTION TO THE POLYMATH PROJECT AND DENSITY HALESJEWETT AND MOSER NUMBERS | 651 |

POLYMATH AND THE DENSITY HALESJEWETT THEOREM | 658 |

DENSITY HALESJEWETT AND MOSER NUMBERS | 689 |

MY EARLY ENCOUNTERS WITH SZEMEREDI | 754 |

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### Common terms and phrases

additive combinatorics algorithm apply argument arithmetic progressions assume bipartite bipartite graph bond percolation combinatorial line complexity Comput conjecture consider constant construction contains Corollary counting lemma critical pair cube curves defined definition denote dense dense graphs density Hales–Jewett theorem Discrete disjoint distribution edges elements Endre Szemerédi Erdös exists finite function f given Gowers graph G graphon grey face Hence holds homomorphism hyperedge hypergraphs implies independent inequality integer k-graph Komlós least Lebesgue measure line-free linear extension log2 lower bound Math Mathematics measure Moser set nilmanifold nilsequence norm Note number theory obtain percolation model permutations points polynomial sequence poset positive primes probability problem proof of Theorem Proposition prove pseudorandom quasirandom random graph rectangle regularity lemma regularity partitions result Rödl Sárközy satisfies Section self-dual slices subgraph subset subspace Suppose symmetry Szemerédi Szemerédi’s theorem Terence Tao translate triangle trivial vector vertex vertices