Szemer 's influence on today's mathematics, especially in combinatorics, additive number theory, and theoretical computer science, is enormous. This volume is a celebration of Szemer '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, Szemer 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. TOC:J. Nesetril, P. Ossona De Mendez Extremal Problems For Sparse Graphs ei-Chu Chang An Estimate of Incomplete Mixed Character Sums ichael A. Nielsen Introduction to the Polymath Project Jozsef Beck Super-Uniformity of the Typical Billiard Path . T. Keller, N. Streib, W. T. Trotter Online Linear Discrepancy Of Partially Ordered Sets Luca Trevisan Pseudorandomness in Computer Science and in Additive Combinatorics Bela Bollobas and Oliver Riordan Percolation on self-dual polygon configurations L. Lovasz, B. Szegedy Regularity Partitions and the Topology on Graphons Noga Alon Universality, Tolerance, Chaos and Order Ben Green And Terence Tao An Arithmetic Regularity Lemma, An Associated Counting Lemma, And Applications Ben Green And Terence Tao Yet another proof of Szemeredi's theorem D.H.J. POLYMATH Density Hales-Jewett And Moser Numbers Imre Ruzsa Towards a noncommutative Plunnecke-type inequality Tim Gowers Polymath And The Density Hales-Jewett Theorem Joel Spencer Quasirandom Multitype Graphs V. Rodl and A. Rucinski Dirac-type questions for hypergraphs Hoi Nguyen and Van Vu Squares In Sumsets Jean Bourgain On Exponential Sums in Finite Fields Y. Kohayakawa, V. Rodl, M. Schacht, J. Skokan On The Triangle Removal Lemma For Subgraphs Of Subgraphs Of Sparse Pseudorandom Graphs Andras Hajnal My early encounters with Szemeredi J. Fox, F. Frati, J. Pach, R. Pinchasi Crossings between curves with many tangancies J. Pintz Are there arbitrarily long arithmetic progressions in the sequence of twin primes? . Loebl, B. Reed, A. Scott, A. Thomason, S. Thomasse Almost all H-free graphs have the Erdos-Hajnal property 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.