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Ks-Free Graphs Without Large Kr-Free Subgraphs

Published online by Cambridge University Press:  12 September 2008

Michael Krivelevich
Michael Krivelevich
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel e-mail: mat2800@technion.technion.ac.il
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Abstract

The main result of this paper is that for every 2 ≤ r < s, and n sufficiently large, there exist graphs of order n, not containing a complete graph on s vertices, in which every relatively not too small subset of vertices spans a complete graph on r vertices. Our results improve on previous results of Bollobás and Hind.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 3 , September 1994 , pp. 349 - 354
Copyright
Copyright © Cambridge University Press 1994

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References

[1]Ajtai, M., Erdős, P., Komlós, J. and Szemerédi, A. (1981) On Turán's theorem for sparse graphs. Combinatorica 1 313–317.CrossRefGoogle Scholar
[2]Alon, N., Spencer, J. and Erdős, P. (1992) The probabilistic method, John Wiley & Sons.Google Scholar
[3]Bollobás, B. and Hind, H. R. (1991) Graphs without large triangle free subgraphs. Discrete Math. 87 119–131.CrossRefGoogle Scholar
[4]Erdős, P. and Gallai, T. (1961) On the minimal number of vertices representing the edges of a graph. Publ. Math. Inst. Hungar. Acad. Sci. 6 181–203.Google Scholar
[5]Erdős, P., Hajnal, A. and Tuza, Z. (1991) Local constraints ensuring small representing sets. J. Combin. Theory, Ser. A 58 78–84.CrossRefGoogle Scholar
[6]Erdős, P. and Lovász, L. (1975) Problems and results on 3-chromatic hypergraphs and some related questions. In: Hajnal, A. et al. (eds.) Infinite and finite sets, North-Holland 609–628.Google Scholar
[7]Erdős, P. and Rogers, C. A. (1962) The construction of certain graphs. Canad. J. Math. 14 702–707.CrossRefGoogle Scholar
[8]Janson, S., Luczak, T. and Ruciński, A. (1990) An exponential bound for the probability of non-existence of a specified graph in a random graph. In: Kazoński, M. et al. (eds.) Random Graphs '87, John Wiley & Sons 73–87.Google Scholar
[9]Linial, N. and Rabinovich, Yu. (1994) Local and global clique numbers. J. Combin. Theory, Ser. B. 61 5–15.CrossRefGoogle Scholar
[10]Tuza, Z. (1989) Minimum number of elements representing a set system of given rank. J. Combin. Theory, Ser. A 52 84–89.CrossRefGoogle Scholar