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OLD AND NEW EUCLIDEAN RAMSEY THEOREMS

(i) (Adam [O].) Any rearrangement of the integers { 1, 2, ..., nz + I} always contains a monotone subsequence of length n + 1. (ii) (van der Waerden [lS].) For any partition of the set Z+ of positive integers into finitely many classes, one of the classes must contain arbitrarily long arithmetic progressions. (iii) (Ramsey [12].) For any partition of the k-element subsets of E+ into finitely many classes, some class must contain all the k-element subsets of some infinite subset X G E+. (iv) (Hindman [9].) For any partition of Z+ into finitely many classes, some class must contain all the (finite) subset sums of some infinite subset X G E+. (v) (Graham et al. [S]; also see [16].) For any integers k, 1, and r and any finite field F, there exists an integer N = N(k, I , r, F ) such that if all the k-dimensional subspaces of an n-dimensional vector space G over F are partitioned into r classes, then some class must contain all the k-dimensional subspaces of some I-dimensional subspace of G.