Abstract
According to Mantel’s theorem, a trianglefree graph on n points has at most n^{2}/4 edges. A linear kgraph is a set of points together with some kelement subsets, called edges, such that any two edges intersect in at most one point. The kgraph F^{k}, called a fan, consists of k edges that pairwise intersect in exactly one point v, plus one more edge intersecting each of these edges in a point different from v. We extend Mantel’s theorem as follows: fanfree linear kgraphs on n points have at most n^{2}/k^{2} edges. This extension nicely illustrates the difficulties of hypergraph Turán problems. The determination of the case of equality leads to transversal designs on n points with k groups—for k = 3 these are equivalent to Latin squares. However, in contrast to the graph case, new structures and open problems emerge when n is not divisible by k.
Original language  English 

Pages (fromto)  263268 
Number of pages  6 
Journal  American Mathematical Monthly 
Volume  127 
Issue number  3 
DOIs 

Publication status  Published  Mar 15 2020 
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Keywords
 05C70
 MSC: Primary 05D05
 Secondary 05B15
ASJC Scopus subject areas
 Mathematics(all)