A contribution to a problem of directed graphs (Q1570888)

The author explains the problem of finding minimum-cardinality tournaments with the property that for any \(k\) vertices of the tournament there is one vertex that beats them all. This problem connected with some problem in logic was encountered by the author in 1934, and later turned out to be a particularly nice example of an application of the probabilistic method. In this paper, the author shows how to determine some small values (the smallest tournament for \(k=3\) has 19 vertices), a lower bound, and a connection to the quadratic residue tournaments. These tournaments already occured in related extremal problems for tournaments where the probabilistic method also gave good constructions.