VC-Dimension of Sets of Permutations

We deene the VC-dimension of a set of permutations A S n to be the maximal k such that there exist distinct i 1 ; :::; i k 2 f1; :::; ng that appear in A in all possible linear orders, that is, every linear order of fi 1 ; :::; i k g is equivalent to the standard order of f(i 1); :::; (i k)g for at least one permutation 2 A. In other words, the VC-dimension of A is the maximal k such that for some i 1 ; :::; i k the restriction of A to fi 1 ; :::; i k g 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 A S n with VC-dimension 2 is of size < C n. This is analogous to Sauer's lemma for the case of VC-dimension 2.