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We set up a general context in which one can prove Sauer–Shelah type lemmas. We apply our general results to answer a question of Bhaskar [1] and give a slight improvement to a result of Malliaris and Terry [7]. We also prove a new Sauer–Shelah type lemma in the context of 
$ \operatorname {\textrm{op}}$
-rank, a notion of Guingona and Hill [4].
