Unavoidable stars in 3-graphs
Denote by f(n,k) the maximal size of a family of 3-subsets of an n-set, not containing k members with the same one-element pairwise intersection (a k-star). Improving earlier bounds by \textit{R. A. Duke} and \textit{P. Erdős} [Proc. 8th Southeast Conf. on Combinatorics, Graph Theory and Computing, Baton Rouge 1977, 247-252 (1977; Zbl 0443.05002)] and \textit{P. Frankl} [Acta Math. Acad. Sci. Hung. 32, 157-160 (1978; Zbl 0407.05013)], the author proves \(k(k-1)(n-(5k+2)/3)\leq f(n,k)\leq k(k-1)n\) for k odd, \(k(k-3/2)(n-k-2)+2k-3\leq f(n,k)\leq(k-1)(k-1/2)n\) for k even. The proof uses a tricky weight-function on the pairs of the underlying set.
scientific article; zbMATH DE number 3855158