Abstract
LetH be a fixed graph of chromatic numberr. It is shown that the number of graphs onn vertices and not containingH as a subgraph is\(2^{(\begin{array}{*{20}c} n \\ 2 \\ \end{array} )(1 - \frac{1}{{r - 1}} + o(1))} \). Leth r (n) denote the maximum number of edges in anr-uniform hypergraph onn vertices and in which the union of any three edges has size greater than 3r − 3. It is shown thath r (n) =o(n 2) although for every fixedc < 2 one has lim n→∞ h r (n)/n c = ∞.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alon, N.: Personal communication 1985
Behrend, F.A.: On sets of integers which contain no three elements in arithmetic progression Proc, Nat. Acad. Sci.,23, 331–332 (1946)
Brown, W.G., P. Erdös, V.T. Sós: Some extremal problems onr-graphs. In: Proc. Third Ann. Arbor Conf. on Graph Th., pp. 53–63. New York: Academic Press 1973
de Caen, D.: Extension of a theorem of Moon and Moser on complete subgraphs. Ars Comb.16, 5–10 (1983)
de Caen, D.: On Turán's hypergraph problem. Ph.D. Thesis, University of Toronto 1982
Erdös, P.: On sequences of integers no one of which divides the product of two others and on related problems. Mitt. Forschung. Inst. Math. Mech. Tomsk,2, 74–82 (1938)
Erdös, P., Stone, A.H.: On the structure of linear graphs. Bull. Amer. Math. Soc.,52, 1087–1091 (1946)
Erdös, P., Kleitman, D.J., Rothschild, B.L.: Asymptotic enumeration ofK n -free graphs. In: International Coll. Comb., Atti dei Convegni Licei 17, vol. 2, pp. 19–27. Rome 1976
Erdös, P., Simonovits, M.: A limit theorem in graph theory. Studia Sci. Mat. Hung. Acad.1, 51–57 (1966)
Erdös, P.: Problems and results in combinatorial analysis. In: Colloq. Internationale sulle Teorie Combinatoria, Rome, 1973, vol. 2, pp. 3–17. Rome: Acad. Naz. Licei 1976
Frankl, P., Fiiredi, Z.: An exact result for 3-graphs. Discrete Math.50, 323–328 (1984)
Frankl, P., Füredi, Z.: Traces of finite sets (in preparation)
Frankl, P., Rödl V.: Lower bounds for Turán's problem. Graphs and Combinatorics1, 27–30 (1985)
Giraud, G.: unpublished (1983)
Kleitman, D.J., Winston, K.J.: The asymptotic number of lattices. In: Combinatorial Mathematics, Optimal Designs and their Applications, edited by Shrivastava. Ann. Discrete Math.6, 243–249 (1980)
Kolaitis, P.H., Prömel, H.J., Rothschild, B.L.:K 1+1-free graphs: asymptotic structure and a 0–1 law (manuscript)
Mantel, W.: Problem 28. Wiskundige Opgaven10, 60–61 (1907)
Rödl, V.: On a packing and covering problem. Europ. J. Comb.5, 69–78 (1985)
Rödl, V.: On universality of graphs with uniformly distributed edges. Discrete Math. (to appear)
Roth, K.F.: On certain sets of integers. J. London Math. Soc.28, 104–109 (1953)
Ruzsa, I.Z., Szemerédi, E.: Triple systems with no six points carrying three triangles. Coll. Math. Soc. Janos Bolyai18, 939–945 (1978)
Szemerédi, E.: Regular partitions of graphs. In: Proc. Colloq. Int. CNRS, pp. 399–401. Paris: CNRS 1976
Turán, P.: An extremal problem in graph theory (in Hungarian). Mat. Fiz. Lapok48, 436–452 (1941)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Erdös, P., Frankl, P. & Rödl, V. The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs and Combinatorics 2, 113–121 (1986). https://doi.org/10.1007/BF01788085
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01788085