The Mathematics of Endre Szemerédi

  • W. T. GowersEmail author
Part of the The Abel Prize book series (AP)


Endre Szemerédi is a towering figure in combinatorics and one of the great mathematicians of the second half of the twentieth century. In this article we discuss some of his most famous results and give a flavour of their proofs.


Bipartite Graph Arithmetic Progression Regularity Lemma Ramsey Number Sorting Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Supplementary material

A lecture by Prof. Gowers in connection with the Abel Prize 2012 to Endre Szemeredi (MP4 295 MB)

A lecture by Prof. Lovasz in connection with the Abel Prize 2012 to Endre Szemeredi (MP4 266 MB)

The Abel Lecture by Endre Szemeredi, the Abel Laureate 2012 (MP4 316 MB)

A lecture by Prof. Wigderson in connection with the Abel Prize 2012 to Endre Szemeredi (MP4 345 MB)


  1. 1.
    Ajtai, M., Komlós, J., Szemerédi, E.: A note on Ramsey numbers. J. Comb. Theory, Ser. A 29, 354–360 (1980) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ajtai, M., Komlós, J., Szemerédi, E.: A dense infinite Sidon sequence. Eur. J. Comb. 2, 1–11 (1981) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ajtai, M., Komlós, J., Szemerédi, E.: An O(nlogn) sorting network. Combinatorica 3, 1–19 (1983) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ajtai, M., Szemerédi, E.: Sets of lattice points that form no squares. Studia Sci. Math. Hung. 9, 9–11 (1975) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bárány, I.: Applications of graph and hypergraph theory in geometry. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry. MSRI Publications, vol. 52, pp. 31–50 (2005) zbMATHGoogle Scholar
  6. 6.
    Batcher, K.: Sorting networds and their applications. AFIPS Spring Joint Comput. Conf. 32, 307–314 (1968) Google Scholar
  7. 7.
    Bourgain, J., Vu, V.H., Wood, P.M.: On the singularity probability of discrete random matrices.
  8. 8.
    Chung, F., Graham, R.L., Wilson, R.M.: Quasi-random graphs. Combinatorica 9, 345–362 (1989) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Elekes, Gy.: On the number of sums and products. Acta Arith. 81, 365–367 (1997) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Erdős, P.: Some remarks on the theory of graphs. Bull. Am. Math. Soc. 53, 292–294 (1947) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Erdős, P., Szemerédi, E.: On sums and products of integers. In: Erdős, P., Alpar, L., Halasz, G. (eds.) Studies in Pure Mathematics: To the Memory of Paul Turán, pp. 213–218. Birkhäuser, Basel (1983) CrossRefGoogle Scholar
  12. 12.
    Erdős, P., Turán, P.: On some sequences of integers. J. Lond. Math. Soc. 11, 261–264 (1936) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Graver, J.E., Yackel, J.: Some graph theoretic results associated with Ramsey’s theorem. J. Comb. Theory 4, 125–175 (1968) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Guth, L., Katz, N.H.: On the Erdős distinct distances problem in the plane.
  15. 15.
    Kahn, J., Komlós, J., Szemerédi, E.: On the probability that a random ±1-matrix is singular. J. Am. Math. Soc. 8, 223–240 (1995) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kim, J.H.: The Ramsey number R(3,t) has order of magnitude t 2/logt. Random Struct. Algorithms 7, 173–207 (1995) CrossRefGoogle Scholar
  17. 17.
    Komlós, J.: On the determinant of (0,1) matrices. Studia Sci. Math. Hung. 2, 387–399 (1968) zbMATHGoogle Scholar
  18. 18.
    Komlós, J., Pintz, J., Szemerédi, E.: A lower bound for Heilbronn’s problem. J. Lond. Math. Soc. 25, 13–24 (1982) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Roth, K.F.: On a problem of Heilbronn. J. Lond. Math. Soc. 26, 198–204 (1951) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Roth, K.F.: On certain sets of integers, I. J. Lond. Math. Soc. 28, 104–109 (1953) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Roth, K.F.: On a problem of Heilbronn, II. Proc. Lond. Math. Soc. 25, 193–212 (1972) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ruzsa, I.Z., Szemerédi, E.: Triple systems with no six points carrying three triangles. In: Combinatorics, Proc. Fifth Hungarian Colloq., Keszthely, 1976. Coll. Math. Soc. J. Bolyai 18, Volume II, pp. 939–945. North Holland, Amsterdam (1978) Google Scholar
  23. 23.
    Schmidt, W.M.: On a problem of Heilbronn. J. Lond. Math. Soc. 4, 545–550 (1972) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Solymosi, J.: Note on a generalization of Roth’s theorem. In: Pach, J. (ed.) Discrete and Computational Geometry. Algorithms Combin., vol. 25, pp. 825–827. Springer, Berlin (2003) CrossRefGoogle Scholar
  25. 25.
    Székely, L.A.: Crossing numbers and hard Erdős problems in discrete geometry. Comb. Probab. Comput. 6, 353–358 (1997) CrossRefGoogle Scholar
  26. 26.
    Szemerédi, E.: On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27, 199–245 (1975) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Szemerédi, E., Trotter, W.T.: Extremal problems in discrete geometry. Combinatorica 3, 381–392 (1983) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tao, T.: Some ingredients in Szemerédi’s proof of Szemerédi’s theorem. Blog post (2012).
  29. 29.
    Tao, T., Vu, V.H.: On random ±1 matrices: singularity and determinant. Random Struct. Algorithms 28, 1–23 (2006) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Tao, T., Vu, V.H.: On the singularity probability of random Bernoulli matrices. J. Am. Math. Soc. 20, 603–628 (2007) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Thomason, A.: Pseudo-random graphs. In: Karoński, M. (ed.) Proceedings of Random Graphs, Poznań, 1985, pp. 307–331. North Holland, Amsterdam (1987) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Royal Society 2010 Anniversary Research ProfessorCentre for Mathematical SciencesCambridgeUK

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