Tournaments, Johnson Graphs, and NC-Teaching
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Quite recently a teaching model, called "No-Clash Teaching" or simply "NC-Teaching", had been suggested that is provably optimal in the following strong sense. First, it satisfies Goldman and Matthias' collusion-freeness condition. Second, the NC-teaching dimension (= NCTD) is smaller than or equal to the teaching dimension with respect to any other collusion-free teaching model. It has also been shown that any concept class which has NC-teaching dimension d and is defined over a domain of size n can have at most 2^d nd concepts. The main results in this paper are as follows. First, we characterize the maximum concept classes of NC-teaching dimension 1 as classes which are induced by tournaments (= complete oriented graphs) in a very natural way. Second, we show that there exists a family (_n)_n≥1 of concept classes such that the well known recursive teaching dimension (= RTD) of _n grows logarithmically in n = |_n| while, for every n≥1, the NC-teaching dimension of _n equals 1. Since the recursive teaching dimension of a finite concept class is generally bounded log||, the family (_n)_n≥1 separates RTD from NCTD in the most striking way. The proof of existence of the family (_n)_n≥1 makes use of the probabilistic method and random tournaments. Third, we improve the afore-mentioned upper bound 2^dnd by a factor of order √(d). The verification of the superior bound makes use of Johnson graphs and maximum subgraphs not containing large narrow cliques.
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