A bipartite Erdős-Ko-Rado theorem (Q2366030)

This paper gives a new proof for the following result: Let \(F_ 1\) and \(F_ 2\) be nonvoid families on an \(n\)-element underlying set containing at most \(k\)-element (at least \((n-k)\)-element, resp.) subsets. Furthermore assume that \(F_ 1\cup F_ 2\) is inclusion free. Then \[ | F_ 1|+| F_ 2|\leq 1+{n\choose k}-{n-k\choose k}. \] (The extremal systems are described, as well.) This result is the well- known Hilton-Milner theorem [Q. J. Math., Oxf. II. Ser. 18, 369-384 (1967; Zbl 0168.262)] in a complemented form.