A Multidimensional Generalization of the Erdős–Szekeres Lemma on Monotone Subsequences

We consider an extension of the Monotone Subsequence lemma of Erdos and Szekeres in higher dimensions. Let v1,…,vn ∈ Rd be a sequence of real vectors. For a subset I ⊆ [n] and vector sc ∈ {0,1}d we say that I is sc-free if there are no i < j ∈ I, such that, for every k = 1,…,d, vik < vik if and only if sck = 0. We construct sequences of vectors with the property that the largest sc-free subset is small for every choice of sc. In particular, for d = 2 the largest sc-free subset is O(nf) for all the four possible sc. The smallest possible value remains far from being determined.We also consider and resolve a simpler variant of the problem.