Schütte's Tournament Problem and Intersecting Families of Sets

Abstract

In this paper we consider a bipartite version of Schütte's well-known tournament problem. A bipartite tournament $T=(A,B,E)$ with teams $A$ and $B$, and set of arcs $E$, has the property $S_{k,l}$ if for any subsets $K\subseteq A$ and $L\subseteq B$, with $|K| =k$ and $| L | =l$, there exist conquerors of $K$ and $L$ in opposite teams. The task is to estimate, for fixed $k$ and $l$, the minimum number $f(k,l)=| A | + | B | $ of players in a tournament satisfying property $S_{k,l}$. We achieve this goal by reformulating the problem in terms of intersecting set families and applying probabilistic as well as constructive methods. Intriguing connections with some famous problems of this area have emerged in this way, leading to new open questions.