On linear combinatorics I. Concurrency—An algebraic approach

Abstract

This article is the first one in a series of three. It contains concurrency results for sets of linear mappings of ℝ with few compositions and/or small image sets. The fine structure of such sets of mappings will be described in part II [3]. Those structure theorems can be considered as a first attempt to find Freiman-Ruzsa type results for a non-Abelian group. Part III [4] contains some geometric applications.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Antal Balog andEndre Szemerédi: A statistical theorem of set addition,Combinatorica,14 (1994), 263–268.

    Google Scholar 

  2. [2]

    József Beck: On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős,Combinatorica,3 (3–4) (1983), 281–297.

    Google Scholar 

  3. [3]

    György Elekes: On linear combinatorics II,Combinatorica, to appear.

  4. [4]

    György Elekes: On linear combinatorics III,Combinatorica, to appear.

  5. [5]

    Gregory A. Freiman:Foundations of a Structural Theory of Set Addition (in Russian), Kazan Gos. Ped. Inst., Kazan, 1966.

    Google Scholar 

  6. [6]

    Gregory A. Freiman:Foundations of a Structural Theory of Set Addition, Translation of Mathematical Monographs vol. 37, Amer. Math. Soc., Providence, R.I., USA, 1973.

    Google Scholar 

  7. [7]

    Tibor Gallai: Solution of problem 4065,Am. Math. Monthly,51 (1944), 169–171.

    Google Scholar 

  8. [8]

    The Mathematics of Paul Erdős, Ron Graham and Jaroslav Nešetřil, eds. Springer, 1996.

  9. [9]

    J. Jackson:Rational Amusements for Winter Evenings. Longman Hurst Rees Orme and Brown, London, 1821.

    Google Scholar 

  10. [10]

    Miklós Laczkovich andImre Z. Ruzsa:The Number of Homothetic Subsets, In: [8] Ron Graham and Jaroslav Nešetřil, eds., Springer, 1996.

  11. [11]

    Imre Z. Ruzsa: Arithmetical progressions and sum sets. Technical Report 91-51, DIMACS Center for Discrete Math. and Theoretical Comp. Sci., Rutgers University, 1991.

  12. [12]

    Imre Z. Ruzsa: Generalized arithmetic progressions and sum sets,Acta Math. Sci. Hung.,65 (1994), 379–388.

    Google Scholar 

  13. [13]

    J. J. Sylvester: Problem 2473.Math. Questions from the Educational Times,8 (1867), 106–107.

    Google Scholar 

  14. [14]

    J. J. Sylvester. Problem 2572.Math. Questions from the Educational Times,45 (1886), 127–128.

    Google Scholar 

  15. [15]

    J. J. Sylvester: Problem 3019.Math. Questions from the Educational Times,45 (1886), 134.

    Google Scholar 

  16. [16]

    J. J. Sylvester: Problem 11851.Math. Questions from the Educational Times,59, (1893), 98–99.

    Google Scholar 

  17. [17]

    Endre Szemerédi: Regular partitions of graphs,Colloques Internationaux C.N.R.S.: Problèmes Combinatoires et Théorie des Graphes, No 260:399–401, 1978.

  18. [18]

    Endre Szemerédi andW. T. Trotter Jr: Extremal problems in Discrete Geometry,Combinatorica,3 (3–4) (1983), 381–392.

    Google Scholar 

  19. [19]

    Vera T. Sós: personal communication.

Download references

Author information

Affiliations

Authors

Additional information

Dedicated to the memory of P. Erdős

Research partially supported by HU-NSF grants OTKA T014302 and T019367.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Elekes, G. On linear combinatorics I. Concurrency—An algebraic approach. Combinatorica 17, 447–458 (1997). https://doi.org/10.1007/BF01194999

Download citation

Mathematics Subject Classification (1991)

  • 52C10
  • 51A25
  • 20F12