An Extension of Mantel's Theorem to k-Graphs.

According to Mantel's theorem, a triangle-free graph on n points has at most n 2 / 4 edges. A linear k-graph is a set of points together with some k-element subsets, called edges, such that any two edges intersect in at most one point. The k-graph Fk, 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: fan-free linear k-graphs 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. [ABSTRACT FROM AUTHOR]