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Constructive Upper Bounds for the Turán Number

Published online by Cambridge University Press:  01 December 1997

Mathematical Institute, University of Heidelberg, Heidelberg, Germany; (e-mail:


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

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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

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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.

Research Article
1997 Cambridge University Press

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