Minimum Tournaments with the Strong $S_k$-Property and Implications for Teaching

TL;DR

This paper introduces a stronger version of the $S_k$-property in tournaments, explores its theoretical properties, and demonstrates its application in teaching models, showing a significant separation between No-Clash and recursive teaching.

Contribution

It defines the strong $S_k$-property, extends existing results to this new notion, and applies it to construct concept classes that separate teaching models by a logarithmic factor.

Findings

Several results on the $S_k$-property extend to the strong $S_k$-property.

An infinite family of concept classes can be taught with one example in the No-Clash model.

A logarithmic separation between No-Clash and recursive teaching models is established.

Abstract

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 $\cC$.