A generalization of Sauerʹs lemma

We generalize Sauerʹs lemma to multivalued functions, proving tight bounds on the cardinality of subsets of ∏i = 1m {0, …, Nm} which avoid certain patterns. In addition, we give an application of this result, bounding the uniform rate of convergence of empirical estimates of the expectations of a set of random variables to their true expectations.