Strictly monotonic multidimensional sequences and stable sets in pillage games

Abstract

Let S ⊂ R n have size | S | > ℓ 2 n − 1 . We show that there are distinct points { x 1 , … , x ℓ + 1 } ⊂ S such that for each i ∈ [ n ] , the coordinate sequence ( x i j ) j = 1 ℓ + 1 is strictly increasing, strictly decreasing, or constant, and that this bound on | S | is best possible. This is analogous to the Erdős–Szekeres theorem on monotonic sequences in R . We apply these results to bound the size of a stable set in a pillage game. We also prove a theorem of independent combinatorial interest. Suppose { a 1 , b 1 , … , a t , b t } is a set of 2 t points in R n such that the set of pairs of points not sharing a coordinate is precisely { { a 1 , b 1 } , … , { a t , b t } } . We show that t ⩽ 2 n − 1 , and that this bound is best possible.