Ramsey theory is a dynamic area of combinatorics that has various applications in analysis, ergodic theory, logic, number theory, probability theory, theoretical computer science, and topological dynamics.

This book is devoted to one of the most important areas of Ramsey theory—the Ramsey theory of product spaces. It is a culmination of a series of recent breakthroughs by the two authors and their students who were able to lift this theory to the infinite-dimensional case. The book presents many major results and methods in the area, such as Szemerédi's regularity method, the hypergraph removal lemma, and the density Hales–Jewett theorem.

This book addresses researchers in combinatorics but also working mathematicians and advanced graduate students who are interested in Ramsey theory. The prerequisites for reading this book are rather minimal: it only requires familiarity, at the graduate level, with probability theory and real analysis. Some familiarity with the basics of Ramsey theory would be beneficial, though not necessary.

I think that this book has a good chance of becoming a classic on density Ramsey theory at the level of the Graham–Rothschild–Spencer book on basic Ramsey theory.

—Stevo Todorcevic, University of Toronto

The book by Dodos and Kanellopoulos is first-rate! It is timely, well written, and has a great selection of topics.

—Ron Graham, University of California, San Diego

Readership

Graduate students and researchers interested in Ramsey theory.

• M. Ajtai and E. Szemerédi, Sets of lattice points that form no squares, Stud. Sci. Math. Hungar. 9 (1974), 9–11.
• N. Alon, R. A. Duke, H. Lefmann, V. Rödl and R. Yuster, The algorithmic aspects of the regularity lemma, J. Algorithms 16 (1994), 80–109.
• N. Alon, E. Fischer, M. Krivelevich and M. Szegedy, Efficient testing of large graphs, Combinatorica 20 (2000), 451–476.
• N. Alon and J. Spencer, The Probabilistic Method, 2nd edition, John Wiley & Sons, 2000.
• S. A. Argyros, V. Felouzis and V. Kanellopoulos, A proof of Halpern–Läuchli partition theorem, Eur. J. Comb. 23 (2002), 1–10.
• T. Austin, Deducing the density Hales–Jewett theorem from an infinitary removal lemma, J. Theor. Probab. 24 (2011), 615–633.
• T. Austin and T. Tao, On the testability and repair of hereditary hypergraph properties, Random Struct. Algorithms 36 (2010), 373–463.
• J. E. Baumgartner, A short proof of Hindman’s theorem, J. Comb. Theory, Ser. A 17 (1974), 384–386.
• F. A. Behrend, On sets of integers which contain no three in arithmetic progression, Proc. Natl. Acad. Sci. USA 23 (1946), 331–332.
• V. Bergelson, “Ergodic Ramsey theory—an update”, in Ergodic Theory of $\mathbb {Z}^d$-Actions, London Mathematical Society Lecture Note Series, Vol. 228, Cambridge University Press, 1996, 1–61.
• V. Bergelson, A. Blass and N. Hindman, Partition theorems for spaces of variable words, Proc. Lond. Math. Soc. 68 (1994), 449–476.
• V. Bergelson and A. Leibman, Set-polynomials and polynomial extension of the Hales–Jewett theorem, Ann. Math. 150 (1999), 33–75.
• A. Blass, A partition theorem for perfect sets, Proc. Amer. Math. Soc. 82 (1981), 271–277.
• A. Blass, Ultrafilters: where topological dynamics $=$ algebra $=$ combinatorics, Topology Proc. 18 (1993), 33–56.
• T. F. Bloom, A quantitative improvement for Roth’s theorem on arithmetic progressions, preprint (2014), available at \verb"arXiv:1405.5800".
• R. Bicker and B. Voigt, Density theorems for finitistic trees, Combinatorica 3 (1983), 305–313.
• P. Billingsley, Probability and Measure, 3rd edition, John Wiley $\&$ Sons, 1995.
• B. Bollobás and V. Nikiforov, “An abstract regularity lemma”, in Building Bridges, Bolyai Society Mathematical Studies, Vol. 19, Springer, 2008, 219–240.
• T. J. Carlson, Some unifying principles in Ramsey theory, Discr. Math. 68 (1988), 117–169.
• T. J. Carlson and S. G. Simpson, A dual form of Ramsey’s theorem, Adv. Math. 53 (1984), 265–290.
• D. Conlon, A new upper bound for diagonal Ramsey numbers, Ann. Math. 170 (2009), 941–960.
• P. Dodos, V. Kanellopoulos and N. Karagiannis, A density version of the Halpern–Läuchli theorem, Adv. Math. 244 (2013), 955–978.
• P. Dodos, V. Kanellopoulos and Th. Karageorgos, Szemerédi’s regularity lemma via martingales, preprint (2014), available at \verb"arXiv:1410.5966".
• P. Dodos, V. Kanellopoulos and K. Tyros, Dense subsets of products of finite trees, Int. Math. Res. Not. 4 (2013), 924–970.
• P. Dodos, V. Kanellopoulos and K. Tyros, A simple proof of the density Hales–Jewett theorem, Int. Math. Res. Not. 12 (2014), 3340–3352.
• P. Dodos, V. Kanellopoulos and K. Tyros, A density version of the Carlson–Simpson theorem, J. Eur. Math. Soc. 16 (2014), 2097–2164.
• P. Dodos, V. Kanellopoulos and K. Tyros, Measurable events indexed by words, J. Comb. Theory, Ser. A 127 (2014), 176–223.
• P. Dodos, V. Kanellopoulos and K. Tyros, A concentration inequality for product spaces, J. Funct. Anal. 270 (2016), 609–620; available at \verb"arXiv:1410.5965".
• G. Elek and B. Szegedy, A measure-theoretic approach to the theory of dense hypergraphs, Adv. Math. 231 (2012), 1731–1772.
• M. Elkin, An improved construction of progression-free sets, Israel J. Math. 184 (2011), 93–128.
• E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symb. Logic 39 (1974), 163–165.
• P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294.
• P. Erdős, “Problems and results on graphs and hypergraphs: similarities and differences”, in Mathematics of Ramsey Theory, Algorithms and Combinatorics, Vol. 5, Springer, 1990, 12–28.
• P. Erdős, P. Frankl and V. Rödl, The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs Comb. 2 (1986), 113–121.
• P. Erdős and A. Hajnal, Some remarks on set theory, IX. Combinatorial problems in measure theory and set theory, Mich. Math. Journal 11 (1964), 107–127.
• P. Erdős and R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc. 2 (1952), 417–439.
• P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compos. Math. 2 (1935), 463–470.
• P. Frankl and V. Rödl, The uniformity lemma for hypergraphs, Graphs Comb. 8 (1992), 309–312.
• P. Frankl and V. Rödl, Extremal problems on set systems, Random Struct. Algorithms 20 (2002), 131–164.
• A. Frieze and R. Kannan, Quick approximation to matrices and applications, Combinatorica 19 (1999), 175–220.
• Z. Füredi, “Extremal hypergraphs and combinatorial geometry”, in Proceedings of the International Congress of Mathematicians, Vol. 2, Birkhäuser, 1995, 1343–1352.
• H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204–256.
• H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Anal. Math. 31 (1978), 275–291.
• H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Anal. Math. 45 (1985), 117–168.
• H. Furstenberg and Y. Katznelson, Idempotents in compact semigroups and Ramsey theory, Israel J. Math. 68 (1989), 257–270.
• H. Furstenberg and Y. Katznelson, A density version of the Hales–Jewett theorem, J. Anal. Math. 57 (1991), 64–119.
• H. Furstenberg and B. Weiss, Markov processes and Ramsey theory for trees, Comb. Prob. Comput. 12 (2003), 547–563.
• F. Galvin, A generalization of Ramsey’s theorem, Notices Amer. Math. Soc. 15 (1968), 548.
• F. Galvin, Partition theorems for the real line, Notices Amer. Math. Soc. 15 (1968), 660.
• F Galvin and K. Prikry, Borel sets and Ramsey’s theorem, J. Symb. Logic 38 (1973), 193–198.
• W. T. Gowers, Lipschitz functions on classical spaces, Eur. J. Comb. 13 (1992), 141–151.
• W. T. Gowers, Lower bounds of tower type for Szemerédi’s uniformity lemma, Geom. Funct. Anal. 7 (1997), 322–337.
• W. T. Gowers, A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), 465–588.
• W. T. Gowers, Quasirandomness, counting and regularity for 3-uniform hypergraphs, Comb. Prob. Comput. 15 (2006), 143–184.
• W. T. Gowers, Hypergraph regularity and the multidimensional Szemerédi theorem, Ann. Math. 166 (2007), 897–946.
• W. T. Gowers, “Erdős and arithmetic progressions”, in Erdős Centennial, Bolyai Society Mathematical Studies, Vol. 25, Springer, 2013, 265–287.
• R. L. Graham, K. Leeb and B. L. Rothschild, Ramsey’s theorem for a class of categories, Adv. Math. 8 (1972), 417–433.
• R. L. Graham and V. Rödl, “Numbers in Ramsey theory”, in Surveys in Combinatorics, 1987, London Mathematical Society Lecture Note Series, Vol. 123, Cambridge University Press, 1987, 111–153.
• R. L. Graham and B. L. Rothschild, Ramsey’s theorem for $n$-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257–292.
• R. L. Graham, B. L. Rothschild and J. H. Spencer, Ramsey Theory, 2nd edition, John Wiley & Sons, 1980.
• B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. Math. 167 (2008), 481–547.
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• N. Hindman, Finite sums from sequences within cells of a partition of $\nn$, J. Comb. Theory, Ser. A 17 (1974), 1–11.
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• N. Alon, R. A. Duke, H. Lefmann, V. Rödl and R. Yuster, The algorithmic aspects of the regularity lemma, J. Algorithms 16 (1994), 80–109.
• N. Alon, E. Fischer, M. Krivelevich and M. Szegedy, Efficient testing of large graphs, Combinatorica 20 (2000), 451–476.
• N. Alon and J. Spencer, The Probabilistic Method, 2nd edition, John Wiley & Sons, 2000.
• S. A. Argyros, V. Felouzis and V. Kanellopoulos, A proof of Halpern–Läuchli partition theorem, Eur. J. Comb. 23 (2002), 1–10.
• T. Austin, Deducing the density Hales–Jewett theorem from an infinitary removal lemma, J. Theor. Probab. 24 (2011), 615–633.
• T. Austin and T. Tao, On the testability and repair of hereditary hypergraph properties, Random Struct. Algorithms 36 (2010), 373–463.
• J. E. Baumgartner, A short proof of Hindman’s theorem, J. Comb. Theory, Ser. A 17 (1974), 384–386.
• F. A. Behrend, On sets of integers which contain no three in arithmetic progression, Proc. Natl. Acad. Sci. USA 23 (1946), 331–332.
• V. Bergelson, “Ergodic Ramsey theory—an update”, in Ergodic Theory of $\mathbb {Z}^d$-Actions, London Mathematical Society Lecture Note Series, Vol. 228, Cambridge University Press, 1996, 1–61.
• V. Bergelson, A. Blass and N. Hindman, Partition theorems for spaces of variable words, Proc. Lond. Math. Soc. 68 (1994), 449–476.
• V. Bergelson and A. Leibman, Set-polynomials and polynomial extension of the Hales–Jewett theorem, Ann. Math. 150 (1999), 33–75.
• A. Blass, A partition theorem for perfect sets, Proc. Amer. Math. Soc. 82 (1981), 271–277.
• A. Blass, Ultrafilters: where topological dynamics $=$ algebra $=$ combinatorics, Topology Proc. 18 (1993), 33–56.
• T. F. Bloom, A quantitative improvement for Roth’s theorem on arithmetic progressions, preprint (2014), available at \verb"arXiv:1405.5800".
• R. Bicker and B. Voigt, Density theorems for finitistic trees, Combinatorica 3 (1983), 305–313.
• P. Billingsley, Probability and Measure, 3rd edition, John Wiley $\&$ Sons, 1995.
• B. Bollobás and V. Nikiforov, “An abstract regularity lemma”, in Building Bridges, Bolyai Society Mathematical Studies, Vol. 19, Springer, 2008, 219–240.
• T. J. Carlson, Some unifying principles in Ramsey theory, Discr. Math. 68 (1988), 117–169.
• T. J. Carlson and S. G. Simpson, A dual form of Ramsey’s theorem, Adv. Math. 53 (1984), 265–290.
• D. Conlon, A new upper bound for diagonal Ramsey numbers, Ann. Math. 170 (2009), 941–960.
• P. Dodos, V. Kanellopoulos and N. Karagiannis, A density version of the Halpern–Läuchli theorem, Adv. Math. 244 (2013), 955–978.
• P. Dodos, V. Kanellopoulos and Th. Karageorgos, Szemerédi’s regularity lemma via martingales, preprint (2014), available at \verb"arXiv:1410.5966".
• P. Dodos, V. Kanellopoulos and K. Tyros, Dense subsets of products of finite trees, Int. Math. Res. Not. 4 (2013), 924–970.
• P. Dodos, V. Kanellopoulos and K. Tyros, A simple proof of the density Hales–Jewett theorem, Int. Math. Res. Not. 12 (2014), 3340–3352.
• P. Dodos, V. Kanellopoulos and K. Tyros, A density version of the Carlson–Simpson theorem, J. Eur. Math. Soc. 16 (2014), 2097–2164.
• P. Dodos, V. Kanellopoulos and K. Tyros, Measurable events indexed by words, J. Comb. Theory, Ser. A 127 (2014), 176–223.
• P. Dodos, V. Kanellopoulos and K. Tyros, A concentration inequality for product spaces, J. Funct. Anal. 270 (2016), 609–620; available at \verb"arXiv:1410.5965".
• G. Elek and B. Szegedy, A measure-theoretic approach to the theory of dense hypergraphs, Adv. Math. 231 (2012), 1731–1772.
• M. Elkin, An improved construction of progression-free sets, Israel J. Math. 184 (2011), 93–128.
• E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symb. Logic 39 (1974), 163–165.
• P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294.
• P. Erdős, “Problems and results on graphs and hypergraphs: similarities and differences”, in Mathematics of Ramsey Theory, Algorithms and Combinatorics, Vol. 5, Springer, 1990, 12–28.
• P. Erdős, P. Frankl and V. Rödl, The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs Comb. 2 (1986), 113–121.
• P. Erdős and A. Hajnal, Some remarks on set theory, IX. Combinatorial problems in measure theory and set theory, Mich. Math. Journal 11 (1964), 107–127.
• P. Erdős and R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc. 2 (1952), 417–439.
• P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compos. Math. 2 (1935), 463–470.
• P. Frankl and V. Rödl, The uniformity lemma for hypergraphs, Graphs Comb. 8 (1992), 309–312.
• P. Frankl and V. Rödl, Extremal problems on set systems, Random Struct. Algorithms 20 (2002), 131–164.
• A. Frieze and R. Kannan, Quick approximation to matrices and applications, Combinatorica 19 (1999), 175–220.
• Z. Füredi, “Extremal hypergraphs and combinatorial geometry”, in Proceedings of the International Congress of Mathematicians, Vol. 2, Birkhäuser, 1995, 1343–1352.
• H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204–256.
• H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Anal. Math. 31 (1978), 275–291.
• H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Anal. Math. 45 (1985), 117–168.
• H. Furstenberg and Y. Katznelson, Idempotents in compact semigroups and Ramsey theory, Israel J. Math. 68 (1989), 257–270.
• H. Furstenberg and Y. Katznelson, A density version of the Hales–Jewett theorem, J. Anal. Math. 57 (1991), 64–119.
• H. Furstenberg and B. Weiss, Markov processes and Ramsey theory for trees, Comb. Prob. Comput. 12 (2003), 547–563.
• F. Galvin, A generalization of Ramsey’s theorem, Notices Amer. Math. Soc. 15 (1968), 548.
• F. Galvin, Partition theorems for the real line, Notices Amer. Math. Soc. 15 (1968), 660.
• F Galvin and K. Prikry, Borel sets and Ramsey’s theorem, J. Symb. Logic 38 (1973), 193–198.
• W. T. Gowers, Lipschitz functions on classical spaces, Eur. J. Comb. 13 (1992), 141–151.
• W. T. Gowers, Lower bounds of tower type for Szemerédi’s uniformity lemma, Geom. Funct. Anal. 7 (1997), 322–337.
• W. T. Gowers, A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), 465–588.
• W. T. Gowers, Quasirandomness, counting and regularity for 3-uniform hypergraphs, Comb. Prob. Comput. 15 (2006), 143–184.
• W. T. Gowers, Hypergraph regularity and the multidimensional Szemerédi theorem, Ann. Math. 166 (2007), 897–946.
• W. T. Gowers, “Erdős and arithmetic progressions”, in Erdős Centennial, Bolyai Society Mathematical Studies, Vol. 25, Springer, 2013, 265–287.
• R. L. Graham, K. Leeb and B. L. Rothschild, Ramsey’s theorem for a class of categories, Adv. Math. 8 (1972), 417–433.
• R. L. Graham and V. Rödl, “Numbers in Ramsey theory”, in Surveys in Combinatorics, 1987, London Mathematical Society Lecture Note Series, Vol. 123, Cambridge University Press, 1987, 111–153.
• R. L. Graham and B. L. Rothschild, Ramsey’s theorem for $n$-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257–292.
• R. L. Graham, B. L. Rothschild and J. H. Spencer, Ramsey Theory, 2nd edition, John Wiley & Sons, 1980.
• B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. Math. 167 (2008), 481–547.
• A. Grzegorczyk, Some classes of recursive functions, Diss. Math. 4 (1953), 1–45.
• A. H. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222–229.
• J. D. Halpern and H. Läuchli, A partition theorem, Trans. Amer. Math. Soc. 124 (1966), 360–367.
• J. D. Halpern and A. Lévy, The boolean prime ideal theorem does not imply the axiom of choice, Proceedings of the Symposium in Pure Mathematics, Vol. 13, part I, American Mathematical Society, Providence, R. I., 1967, 83–134.
• N. Hindman, Finite sums from sequences within cells of a partition of $\nn$, J. Comb. Theory, Ser. A 17 (1974), 1–11.
• N. Hindman and R. McCutcheon, Partition theorems for left and right variable words, Combinatorica 24 (2004), 271–286.
• N. Hindman and R. McCutcheon, One sided ideals and Carlson’s theorem, Proc. Amer. Math. Soc. 130 (2002), 2559–2567.
• N. Hindman and D. Strauss, Algebra in the Stone–Čech compactification: theory and applications, Walter de Gruyter, Berlin, 1998.
• V. Kanellopoulos, A proof of W. T. Gowers’ $c_0$ theorem, Proc. Amer. Math. Soc. 132 (2004), 3231–3242.
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