VC-dimension of subsets of Hamming graphs

Following recent work on the VC-dimension of subsets of various pseudorandom graphs, we study the VC-dimension of Hamming graphs, which have proved somewhat resistant to the standard techniques in the literature. Our methods are elementary, and agree with or improve upon previously known results. In particular, for $H(2,q)$ we show tight bounds on the size of a subset of vertices to guarantee VC-dimension 2 or 3. We also prove an assortment of results for other parameters, with many of these being tight as well.