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Graphs and Combinatorics

, Volume 8, Issue 4, pp 299–308 | Cite as

Ramsey problem on multiplicities of complete subgraphs in nearly quasirandom graphs

  • F. Franek
  • V. Rödl
Original Papers

Abstract

Letkt(G) be the number of cliques of ordert in the graphG. For a graphG withn vertices let\(c_t (G) = \frac{{k_t (G) + k_t (\bar G)}}{{\left( {\begin{array}{*{20}c} n \\ t \\ \end{array} } \right)}}\). Letct(n)=Min{ct(G)∶∇G∇=n} and let\(c_t = \mathop {\lim }\limits_{n \to \infty } c_t (n)\). An old conjecture of Erdös [2], related to Ramsey's theorem states thatct=21-(t/2). Recently it was shown to be false by A. Thomason [12]. It is known thatct(G)≈21-(t/2) wheneverG is a pseudorandom graph. Pseudorandom graphs — the graphs “which behave like random graphs” — were inroduced and studied in [1] and [13]. The aim of this paper is to show that fort=4,ct(G)≥21-(t/2) ifG is a graph arising from pseudorandom by a small perturbation.

Keywords

Small Perturbation Random Graph Theorem State Complete Subgraph Ramsey Problem 
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.

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References

  1. 1.
    Chung, F.R.K., Graham, R.L., Wilson, R.M.:Quasi-random Graphs, Combinatorica9, no. 4, 345–362 (1989)MathSciNetGoogle Scholar
  2. 2.
    Erdös, P.:On the number of complete subgraphs contained in certain graphs, Publ. Math. Inst. Hung. Acad. Sci., VII, ser. A3, 459–464 (1962)Google Scholar
  3. 3.
    Erdös, P., Moon, J.W.:On subgroups on the complete bipartite graph, Canad. Math. Bull.7, 35–39 (1964)MathSciNetGoogle Scholar
  4. 4.
    Franek, F., Rödl, V.:2-colorings of complete graphs with small number of monochromatic K 4 subgraphs, to appear in Discr. Math.Google Scholar
  5. 5.
    Frankl, P., Rödl, V., Wilson, R.M.:The number of submatrices of given type in a Hadamard matrix and related results, J. Comb. Theory,44, 317–328 (1988)Google Scholar
  6. 6.
    Goodman, A.W.:On sets of acquaintances and strangers at any party, Amer. Math. Monthly,66, 778–783 (1959)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Giraud, G.:Sur le probleme de Goodman pour les quadrangles et la majoration des nombres de Ramsey, J. Comb. Theory Ser. B30, 237–253 (1979)MathSciNetGoogle Scholar
  8. 8.
    Graham, R.L., Spencer, J.H.:A constructive solution to a tournament problem, Canad. Math. Bull.14, 45–48 (1971)MathSciNetGoogle Scholar
  9. 9.
    Rödl, V.:On universality of graphs with uniformly distributed edges, Discr. Math.59, no. 1-2, 125–134 (1986)zbMATHGoogle Scholar
  10. 10.
    Szemerédi, E.:Regular partitions of graphs, inProc. Colloque Internat. CNRS (J.-C. Bermond et. al., eds.), Paris, 1978, 399–401Google Scholar
  11. 11.
    Sidorenko, A.F.:Tsikly v grafakh i funktional'nye neravenstva, Matematicheskie Zametki,46, no. 5, 72–79 (1989) (in Russian)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Thomason, A.:A disproof of a conjecture of Erdös in Ramsey theory, J. London Math. Soc. (2),39, no. 2, 246–255 (1898)MathSciNetGoogle Scholar
  13. 13.
    Thomason, A.:Pseudo-random graphs, in “Proceedings of Random Graphs, Poznan, '85”, (M. Karonski, ed.), Discrete Applied Math., 307–331.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • F. Franek
    • 1
  • V. Rödl
    • 2
  1. 1.Department of Computer Science and SystemsMcMaster UniversityHamiltonCanada
  2. 2.Departments of Mathematics and Computer Science Emory UniversityAtlantaUSA

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