Separating systems and oriented graphs of diameter two

Authors:
Béla Bollobás;Alex Scott
Affiliations:
Trinity College, Cambridge CB2 1TQ, UK and Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA;Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK
Venue:
Journal of Combinatorial Theory Series B
Year:
2007

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Abstract

We prove results on the size of weakly and strongly separating set systems and matrices, and on cross-intersecting systems. As a consequence, we improve on a result of Katona and Szemeredi [G. Katona, E. Szemeredi, On a problem of graph theory, Studia Sci. Math. Hungar. 2 (1967) 23-28], who proved that the minimal number of edges in an oriented graph of order n with diameter 2 is at least (n/2)log"2(n/2). We show that the minimum is (1+o(1))nlog"2n.