Graphs and Combinatorics

, Volume 11, Issue 2, pp 179–199 | Cite as

What we know and what we do not know about Turán numbers

  • Alexander Sidorenko


The numbers which are traditionally named in honor of Paul Turán were introduced by him as a generalization of a problem he solved in 1941. The general problem of Turán having anextremely simple formulation but beingextremely hard to solve, has become one of the most fascinatingextremal problems in combinatorics. We describe the present situation and list conjectures which are not so hopeless.


Simple Formulation General Problem Present Situation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boyer, E.D., Kreher, D.L., Radziszowski, S.P., Zou, J., and Sidorenko, A.: On (n, 5, 3)-Turán systems. Ars Combinatoria37, 1–19 (1994)Google Scholar
  2. 2.
    Brown, W.G.: On an open problem of Paul Turán concerning 3-graphs. In: Studies in pure mathematics, pp. 91–93. Basel-Boston, Mass.: Birkhäuser 1983Google Scholar
  3. 3.
    Caen, D. de: Extension of a theorem of Moon and Moser on complete subgraphs. Ars Combinatoria16, 5–10 (1983)Google Scholar
  4. 4.
    Caen, D. de: A note on the probabilistic approach to Turán's problem. J. Comb. TheoryB 34, 340–349 (1983)Google Scholar
  5. 5.
    Caen, D. de: The current status of Turán's problem on hypergraphs. In: Extremal Problems for Finite Sets, Visegrád (Hungary), 1991, Bolyai Society Mathematical Studies, Vol. 3, pp. 187–197Google Scholar
  6. 6.
    Caen, D. de, Kreher, D.L., Radziszowski, S.P., and Mills, W.H.: On the coverings oft-sets with (t + 1)-sets:C(9, 5, 4) andC(10, 6, 5). Discrete Mathematics92, 65–77 (1991)CrossRefGoogle Scholar
  7. 7.
    Caen, D. de, Kreher, D.L., and Wiseman, J.: On constructive upper bounds for the Turán numbersT(n, 2r + 1, r). Congressus Numerantium65, 277–280 (1988)Google Scholar
  8. 8.
    Droesbeke, F., Lorea, M.: Détermination de valeurs du nombre de TuránT(n, 4, 6). Cahiers du Cent. Étud. Rech. Oper.24, 185–191 (1982)Google Scholar
  9. 9.
    Erdos, P.: On the combinatorial problems I would most like to see solved. Combinatorica1, 25–42 (1981)Google Scholar
  10. 10.
    Fon-Der-Flaass, D.G.: A method for constructing (3,4)-graphs. Math. Notes44, (1988)Google Scholar
  11. 11.
    Fort, M.K., Hedlund, G.A.: Minimal coverings of pairs by triples. Pacific J of Mathematics8, 709–719 (1958)Google Scholar
  12. 12.
    Frankl, P., Rödl, V.: Lower bounds for Turán's problem. Graphs and Combinatorics1, 213–216 (1985)Google Scholar
  13. 13.
    Füredi, Z.: Covering pairs byq 2 +q + 1 sets. J. Comb. TheoryA 54, 248–271 (1990)CrossRefGoogle Scholar
  14. 14.
    Gardner, B.: Results on coverings of pairs with special reference to coverings by quintuples. Congressus Numerantium23, 169–178 (1971)Google Scholar
  15. 15.
    Giraud, G.: Majoration du nombre de Ramsey ternaires-bicolore en (4,4). C. R. Acad. Sci. ParisAB269, A620-A622 (1969)Google Scholar
  16. 16.
    Giraud, G.: Remarques sur deus problèmes estrémaux. Discrete Mathematics84, 319–321 (1990)CrossRefGoogle Scholar
  17. 17.
    Giraud, G.: Une minoration de la premiere fonction ternaire de Turán, manuscriptGoogle Scholar
  18. 18.
    Gordon, D.M., Kuperberg, G., and Patashnik, O.: New constructions for covering designs. J. Comb. Design, to be publishedGoogle Scholar
  19. 19.
    Katona, G., Nemetz, T., and Simonovits, M.: On a graph problem of Turán, (in Hungarian). Mat. Lapok15, 228–238 (1964)Google Scholar
  20. 20.
    Kim, K.H., Roush, F.W.: On a problem of Turán. In: Studies in pure mathematics, pp. 423–425. Basel-Boston, Mass.: Birkhäuser 1983Google Scholar
  21. 21.
    Kostochka, A.V.: A class of constructions for Turán's (3,4)-problem. Combinatorica2, 187–192 (1982)Google Scholar
  22. 22.
    Kuzyurin, N.N.: Asymptotic investigation of the covering problem. Cybernetic Problems37, 19–56 (1980)Google Scholar
  23. 23.
    Feng-Chu Lai, Chang, G.J.: An upper bound for the transversal numbers of 4-uniform hypergraphs. J. Comb. TheoryB 50, 129–133 (1990)CrossRefGoogle Scholar
  24. 24.
    Lamken, E., Mills, W. H., Mullin, R.C., and Vanstone, S.A.: Covering of pairs by quintuples. J. Comb. TheoryA 44, 49–68 (1987)CrossRefGoogle Scholar
  25. 25.
    Lorea, M., On Turán hypergraphs. Discrete Mathematics22, 281–285 (1978)CrossRefGoogle Scholar
  26. 26.
    Mantel, W.: Vraagstuk XXVIII. Wiskundige Opgaven met de Oplossingen10, 60–61 (1907)Google Scholar
  27. 27.
    Mills, W.H.: On the covering of pairs by quadruples I. J. Comb. TheoryA 13, 55–78 (1972)CrossRefGoogle Scholar
  28. 28.
    Mills, W.H.: On the covering of pairs by quadruples II. J. Comb. TheoryA 13, 138–166 (1973)Google Scholar
  29. 29.
    Mills, W.H.: On the covering of triples by quadruples. In: Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing, Boca Raton, Fla., 1974, pp. 573–581. Winnipeg: Utilitas 1974Google Scholar
  30. 30.
    Mills, W.H.: Covering designs I: covering by a small number of subsets. Ars Combinatoria8, 199–315 (1979)Google Scholar
  31. 31.
    Mills, W.H.: A covering of pairs by quintuples. Ars Combinatoria18, 21–31 (1984)Google Scholar
  32. 32.
    Mills, W.H., Mullin, R.C.: Covering triples by quadruples: an asymptotic solution. J. Comb. TheoryA 41, 117–138 (1986)CrossRefGoogle Scholar
  33. 33.
    Mills, W.H., Mullin, R.C.: Covering pairs by quintuples: the casev ≡ 3 modulo 4. J. Comb. TheoryA 49, 308–322 (1988)CrossRefGoogle Scholar
  34. 34.
    Mills, W.H., Mullin, R.C.: Coverings and packings. In: Contemporary Design Theory: A Collection of Surveys, pp. 371–399. Wiley 1992Google Scholar
  35. 35.
    Mullin, R.C.: On covering pairs by quintuples: the casesv ≡ 3 or 11 modulo 20, J. of Combinatorial Mathematics and Combinatorial Computing2, 133–146 (1987)Google Scholar
  36. 36.
    Mullin, R.C.: On the determination of the covering numbersC(2, 5, v). J. of Combinatorial Mathematics and Combinatorial Computing4, 123–132 (1988)Google Scholar
  37. 37.
    Radziszowski, S.P., Zou, J.: Personal communicationGoogle Scholar
  38. 38.
    Rödl, V.: On a packing and covering problem Europ. J. Comb.5, 69–78 (1985)Google Scholar
  39. 39.
    Schönheim, J.: On coverings. Pacific J. of Mathematics14, 1405–1411 (1964)Google Scholar
  40. 40.
    Sidorenko, A.F.: On Turán numbersT(n, 5, 4) and numbers of monochromatic 4-cliques in 2-colored 3-graphs (in Russian). Voprosi Kibernetiki no. 64, 117–124 (1980)Google Scholar
  41. 41.
    Sidorenko, A.F.: Systems of sets that have theT-property. Moscow University Mathematics Bulletin36, no. 5, 22–26 (1981)Google Scholar
  42. 42.
    Sidorenko, A.F.: The method of quadratic forms and Turán's combinatorial problem. Moscow University Mathematics Bulletin37, no. 1, 1–5 (1982)Google Scholar
  43. 43.
    Sidorenko, A.F. Extremal constants and inequalities for distributions of sums of random vectors (in Russian). Ph.D. Thesis, Moscow State University, 1982Google Scholar
  44. 44.
    Sidorenko, A.F.: The Turán problem for 3-graphs (in Russian). Combinatorial Analysis (Russian), no. 6, pp. 51–57. Moscow State University, 1983Google Scholar
  45. 45.
    Sidorenko, A.F.: Exact values of Turán numbers. Math. Notes42, 913–918 (1987)Google Scholar
  46. 46.
    Sidorenko, A.: Upper bounds on Turán numbers. Submitted to J. Comb. Theory, ser. AGoogle Scholar
  47. 47.
    Stanton, R.G., Bate, J.A.: A computer search for B-coverings. Lect. Notes in Math., no. 829, 37–50 (1980)Google Scholar
  48. 48.
    Stanton, R.G., Buskens, R.W., and Allston, J.L.: Computation of the covering numberN(2, 5, 24). Utilitas Mathematica37, 127–144 (1990)Google Scholar
  49. 49.
    Surányi, J.: Some combinatorial problems of geometry (in Hungarian). Mat. Lapok22, 215–230 (1971)Google Scholar
  50. 50.
    Todorov D.T., Tonchev, V.D.: On some coverings of triples. C. R. Bulg. Acad. Sci.35, 1209–1211 (1982)Google Scholar
  51. 51.
    Todorov, D.T.: A method of constructing coverings. Math. Notes35, 869–876 (1984)Google Scholar
  52. 52.
    Todorov D.T.: Some coverings derived from finite planes. In: Colloq. Math. Soc. János Bolyai, Vol. 37, Finite and Infinite Sets, Eger, 1981, pp. 697–710. Budapest: Akad. Kiadó 1985Google Scholar
  53. 53.
    Todorov, D.T.: On the covering of pairs by 13 blocks. C. R. Bulg. Acad. Sci.38, 691–694 (1985)Google Scholar
  54. 54.
    Todorov, D.T.: On the covering of triples by eight blocks. Serdica12, 20–29 (1986)Google Scholar
  55. 55.
    Todorov, D.T.: Lower bounds for coverings of pairs by large blocks. Combinatorica9, 217–225 (1989)Google Scholar
  56. 56.
    Turán, P.: Egy gráfelméleti szélsöértékfeladatról. Mat. és Fiz. Lapok48, 436–453 (1941)Google Scholar
  57. 57.
    Turán, P.: Research problems. Maguar Tud. Akad. Mat. Kutato Int. Közl.6, 417–423 (1961)Google Scholar
  58. 58.
    Turán, P.: Applications of graph theory to geometry and potential theory. In: Combinatorial Structures and Their Applications, pp. 423–434. New York: Gordon and Breach 1970Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Alexander Sidorenko
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations