Triple Systems Not Containing a Fano Configuration

Authors:
Zoltán Füredi;Miklós Simonovits
Affiliations:
Rényi Institute of Mathematics of the Hungarian Academy of Sciences Budapest, PO Box 127, Hungary-1364 and Department of Mathematics, University of Illinois at Urbana-Champaign Urbana, IL6180 ...;Rényi Institute of Mathematics of the Hungarian Academy of Sciences Budapest, PO Box 127, Hungary-1364 (e-mail: miki@renyi.hu)
Venue:
Combinatorics, Probability and Computing
Year:
2005

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Abstract

A Fano configuration is the hypergraph of 7 vertices and 7 triplets defined by the points and lines of the finite projective plane of order 2. Proving a conjecture of T. Sós, the largest triple system on $n$ vertices containing no Fano configuration is determined (for $n n_1$). It is 2-chromatic with $\binom{n}{3}-\binom{\lfloor n/2 \rfloor}{3} -\binom{\lceil n/2 \rceil}{3}$ triples. This is one of the very few nontrivial exact results for hypergraph extremal problems.