## VC-dimension

of a set of permutations \(\) to be the maximal *k* such that there exist distinct \(\) that appear in *A* in all possible linear orders, that is, every linear order of \(\) is equivalent to the standard order of \(\) for at least one permutation \(\).

In other words, the VC-dimension of *A* is the maximal *k* such that for some \(\) the restriction of *A* to \(\) contains all possible linear orders. This is analogous to the VC-dimension of a set of strings.

Our main result is that there exists a universal constant *C* such that any set of permutations \(\) with VC-dimension 2 is of size \(\). This is analogous to Sauer's lemma for the case of VC-dimension 2.

One corollary of our main result is that any acyclic set of linear orders of \(\) is of size \(\), (a set *A* of linear orders on \(\) is called *acyclic* if no 3 elements \(\) appear in *A* in all 3 orders (*i*,*j*,*k*), (*k*,*i*,*j*) and (*j*,*k*,*i*)). The size of the largest acyclic set of linear orders has interested researchers for many years because it is the largest number of linear orders of *n* alternatives such that the following is always satisfied: if each one of a set of voters chooses one of these orders as his preference then the majority relation between each two alternatives is transitive.

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