This paper establishes an upper bound on the size of a concept class with given recursive teaching dimension (RTD, a teaching complexity parameter.) The upper bound coincides with Sauer’s well-known bound on classes with a fixed VC-dimension. Our result thus supports the recently emerging conjecture that the combinatorics of VC-dimension and those of teaching complexity are intrinsically interlinked.

This paper studies labeled sample compression for multi-label concept classes. For a specific extension of the notion of VC-dimension to multi-label classes, we prove that every maximum multilabel class of dimensiond has a sample compression scheme in which every sample is compressed to a subset of size at most d. We further show that every multi-label class of dimension 1 has a sample compression scheme using only sets of size at most 1. As opposed to the binary case, the latter result is not immediately implied by the former, since there are multi-label concept classes of dimension 1 that are not contained in maximum classes of dimension 1.

This paper studies sample compression of maximum multi-label concept classes for various notions of VC-dimension. It formulates a sufficient condition for a notion of VC-dimension to yield labeled compression schemes for maximum classes of dimension d in which the compression sets have size at most d. The same condition also yields a so-called tight sample compression scheme, which we define to generalize Kuzmin and Warmuth’s unlabeled binary scheme to the multi-label case. The well-known Graph dimension satisfies our sufficient condition, while neither Pollard’s pseudo-dimension nor the Natarajan dimension does.