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On a question of Sós about 3-uniform friendship hypergraphs

✍ Scribed by Stephen G. Hartke; Jennifer Vandenbussche

Publisher: John Wiley and Sons
Year: 2008
Tongue: English
Weight: 130 KB
Volume: 16
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.20183

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✦ Synopsis

Abstract

The well‐known Friendship Theorem states that if G is a graph in which every pair of vertices has exactly one common neighbor, then G has a single vertex joined to all others (a “universal friend”). V. Sós defined an analogous friendship property for 3‐uniform hypergraphs, and gave a construction satisfying the friendship property that has a universal friend. We present new 3‐uniform hypergraphs on 8, 16, and 32 vertices that satisfy the friendship property without containing a universal friend. We also prove that if n ≤ 10 and n ≠ 8, then there are no friendship hypergraphs on n vertices without a universal friend. These results were obtained by computer search using integer programming. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 253–261, 2008

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