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Constructive Upper Bounds for the Turán Number

Published online by Cambridge University Press:  01 December 1997

ULRICH MATTHIAS
Affiliation:
Mathematical Institute, University of Heidelberg, Heidelberg, Germany; (e-mail: umatthia@ix.urz.uni-heidelberg.de)

Abstract

The Turán Number T(n, k, r) is the smallest possible number of edges in a k-graph of order n such that every set of r vertices contains an edge. The limit

formula here

exists, but there is no pair (k, r) with r>k[ges ]3 for which this function could be determined as yet. We give a constructive proof of the upper bound

formula here

for every k and r with r[ges ]k[ges ]2. In the case k=6, r=11 we improve this result, refuting thereby a conjecture of Turán.

Type
Research Article
Copyright
1997 Cambridge University Press

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