Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T23:50:52.846Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimum Number of k-Cliques in Graphs with Bounded Independence Number

Published online by Cambridge University Press:  01 October 2013

OLEG PIKHURKO  and
EMIL R. VAUGHAN
OLEG PIKHURKO
Affiliation:
Mathematics Institute and DIMAP, University of Warwick, Coventry CV4 7AL, UK (e-mail: Pikhurko@warwick.ac.uk)
EMIL R. VAUGHAN
Affiliation:
Centre for Discrete Mathematics, Queen Mary, University of London, London E1 4NS, UK (e-mail: e.vaughan@qmul.ac.uk)
Get access Rights & Permissions [Opens in a new window]

Abstract

Erdős asked in 1962 about the value of f(n,k,l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k,l)=(3,3) was solved by Lorden. Here we solve the problem (for all large n) for (3,l) with 4 ≤ l ≤ 7 and (k,3) with 4 ≤ k ≤ 7. Independently, Das, Huang, Ma, Naves and Sudakov resolved the cases (k,l)=(3,4) and (4,3).

Keywords

Primary 05C35Secondary 90C35
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 6 , November 2013 , pp. 910 - 934
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Supported by the European Research Council (grant agreement no. 306493) and the National Science Foundation of the USA (grant DMS-1100215).

References

[1]Alon, N., Fischer, E., Krivelevich, M. and Szegedy, M. (2000) Efficient testing of large graphs. Combinatorica 20 451–476.Google Scholar
[2]Baber, R. and Talbot, J. (2011) Hypergraphs do jump. Combin. Probab. Comput. 20 161–171.Google Scholar
[3]Bollobás, B. (1978) Extremal Graph Theory, Academic Press.Google Scholar
[4]Cummings, J., Král', D., Pfender, F., Sperfeld, K., Treglown, A. and Young, M. (2013) Monochromatic triangles in three-coloured graphs. J. Combin. Theory Ser. B 103 489–503.Google Scholar
[5]Das, S., Huang, H., Ma, J., Naves, H. and Sudakov, B. (2013) A problem of Erdős on the minimum number of k-cliques. J. Combin. Theory Ser. B 103 344–373.Google Scholar
[6]Erdős, P. (1962) On the number of complete subgraphs contained in certain graphs. Magyar Tud. Akad. Mat. Kutató Int. Kőzl. 7 459–464.Google Scholar
[7]Falgas-Ravry, V., Marchant, E., Pikhurko, O. and Vaughan, E. R. (2013) The codegree threshold for 3-graphs with independent neighbourhoods. E-print arXiv.org:1307.0075.Google Scholar
[8]Falgas-Ravry, V. and Vaughan, E. R. (2012) Turán H-densities for 3-graphs. Electron. J. Combin. 19 P40.Google Scholar
[9]Falgas-Ravry, V. and Vaughan, E. R. (2013) Applications of the semi-definite method to the Turán density problem for 3-graphs. Combin. Probab. Comput. 22 21–54.Google Scholar
[10]Goodman, A. W. (1959) On sets of acquaintances and strangers at any party. Amer. Math. Monthly 66 778–783.Google Scholar
[11]Hatami, H., Hladký, J., Král', D., Norine, S. and Razborov, A. (2013) On the number of pentagons in triangle-free graphs. J. Combin. Theory Ser. A 120 722–732.Google Scholar
[12]Hirst, J. (2011) The inducibility of graphs on four vertices. E-print arXiv.org:1109.1592.Google Scholar
[13]Kalbfleisch, J. G. and Stanton, R. G. (1968) On the maximal triangle-free edge-chromatic graphs in three colors. J. Combin. Theory 5 9–20.CrossRefGoogle Scholar
[14]Lorden, G. (1962) Blue-empty chromatic graphs. Amer. Math. Monthly 69 114–120.Google Scholar
[15]Nikiforov, V. (2001) On the minimum number of k-cliques in graphs with restricted number of independence. Combin. Probab. Comput. 10 361–366.CrossRefGoogle Scholar
[16]Nikiforov, V. (2005) The minimum number of 4-cliques in a graph with triangle-free complement. E-print arXiv.org:math/050121.Google Scholar
[17]Pikhurko, O. (2011) The minimum size of 3-graphs without four vertices spanning no or exactly three edges. Europ. J. Combin. 23 1142–1155.Google Scholar
[18]Pikhurko, O. and Vaughan, E. R. (2013) Minimum number of k-cliques in graphs with bounded independence number. E-print arXiv.1203.4393, version 4.Google Scholar
[19]Ramsey, F. P. (1930) On a problem of formal logic. Proc. London Math. Soc. 30 264–286.Google Scholar
[20]Razborov, A. (2007) Flag algebras. J. Symb. Logic 72 1239–1282.Google Scholar
[21]Razborov, A. (2010) On 3-hypergraphs with forbidden 4-vertex configurations. SIAM J. Discrete Math. 24 946–963.Google Scholar
[22]Thomason, A. (2002) The simplest case of Ramsey's theorem. In Paul Erdős and his Mathematics, Vol. 11 of Bolyai Soc. Math. Studies, Springer, pp. 667–695.Google Scholar
[23]Vaughan, E. R. (2013) Flagmatic: A tool for researchers in extremal graph theory. Version 2.0, http://flagmatic.org/.Google Scholar