  A tournament is said to have the $S_k$-property if, for any set of $k$
players, there is another player who beats them all. Minimum tournaments having
this property have been explored very well in the 1960's and the early 1970's.
In this paper, we define a strengthening of the $S_k$-property that we name
"strong $S_k$-property". We show, first, that several basic results on the
weaker notion remain valid for the stronger notion (and the corresponding
modification of the proofs requires only little extra-effort). Second, it is
demonstrated that the stronger notion has applications in the area of Teaching.
Specifically, we present an infinite family of concept classes all of which can
be taught with a single example in the No-Clash model of teaching while, in
order to teach a class $\cC$ of this family in the recursive model of teaching,
order of $\log|\cC|$ many examples are required. This is the first paper that
presents a concrete and easily constructible family of concept classes which
separates the No-Clash from the recursive model of teaching by more than a
constant factor. The separation by a logarithmic factor is remarkable because
the recursive teaching dimension is known to be bounded by $\log |\cC|$ for any
concept class $\cC$.
