The No-Clash Teaching Dimension is Bounded by VC Dimension

In the realm of machine learning theory, to prevent unnatural coding schemes between teacher and learner, No-Clash Teaching Dimension was introduced as provably optimal complexity measure for collusion-free teaching. However, whether No-Clash Teaching Dimension is upper-bounded by Vapnik-Chervonenkis dimension remains unknown. In this paper, for any finite concept class, we construct fragments of size equals to its Vapnik-Chervonenkis dimension which identify concepts through an ordered compression scheme. Naturally, these fragments are used as teaching sets, one can easily see that they satisfy the non-clashing condition, i.e., this open question is resolved for finite concept classes.