Monotone Subsequences in Any Dimension
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We exhibit sequences ofnpoints inddimensions with no long monotone subsequences, by which we mean when projected in a general direction, our sequence has no monotone subsequences of lengthn+dor more. Previous work proved that this function ofnwould lie betweennand 2n; this paper establishes that the coefficient ofnis one. This resolves the question of how the Erdos?Szekeres result that a (one-dimensional) sequence has monotone subsequences of at mostngeneralizes to higher dimensions.
[1] James B. Shearer,et al. Monotonic subsequences in dimensions higher than one , 1997, Electron. J. Comb..