Monotone Subsequences in Any Dimension

AbstractWe 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 Erdős–Szekeres result that a (one-dimensional) sequence has monotone subsequences of at mostngeneralizes to higher dimensions

Monotonic Subsequences in Dimensions Higher Than One

The 1935 result of Erdos and Szekeres that any sequence of +1 real numbers contains a monotonic subsequence of n + 1 terms has stimulated extensive further research, including a paper of J. B. Kruskal that defined an extension of monotonicity for higher dimensions. This paper provides a proof of a weakened form of Kruskal's conjecture for 2-dimensional Euclidean space by showing that there exist sequences of n points in the plane for which the longest monotonic subsequences have length + 3. Weaker results are obtained for higher dimensions. When points are selected at random from reasonable distributions, the average length of the longest monotonic subsequence is shown to be 2n ##for each dimension