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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)
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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 

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Footnotes

†

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

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