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Endre 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 his achievements and personality, on the occasion of his 70th birthday.

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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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Universality, Tolerance, Chaos and Order.- Super-Uniformity of The Typical Billiard Path.- Percolation on Self-Dual Polygon Configurations.- On Exponential Sums in Finite Fields.- An Estimate of Incomplete Mixed Character Sums.- Crossings Between Curves With Many Tangencies.- An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications.- Yet Another Proof Of Szemerédi´s Theorem.- Online Linear Discrepancy of Partially Ordered Sets.- On The Triangle Removal Lemma For Subgraphs of Sparse Pseudorandom Graphs.- Almost All F-Free Graphs Have The Erdös-Hajnal Property.- Regularity Partitions and The Topology of Graphons.- Extremal Problems for Sparse Graphs.- Squares In Sumsets.- Are There Arbitrarily Long Arithmetic Progressions In The Sequence of Twin Primes?.- Dirac-Type Questions For Hypergraphs - A Survey (Or More Problems For Endre To Solve).- Towards A Noncommutative Plünnecke-Type Inequality.- Quasirandom Multitype Graphs.- Pseudorandomness In Computer Science and In Additive Combinatorics.- To The Polymath Project and "Density Hales-Jewett and Moser Numbers".- Polymath and The Density Hales-Jewett Theorem.- Density Hales-Jewett and Moser Numbers.- My Early Encounters With Szemerédi.

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