Cross-monotone subsequences

Abstract

Two finite real sequences (a 1,...,a k ) and (b 1,...,b k ) are cross-monotone if each is nondecreasing anda i+1−a i ≥b i+1−b i for alli<k. A sequence (α1,..., α n ) of nondecreasing reals is in class CM(k) if it has disjointk-term subsequences that are cross-monotone. The paper shows thatf(k), the smallestn such that every nondecreasing (α1,..., α n ) is in CM(k), is bounded between aboutk 2/4 andk 2/2. It also shows thatg(k), the smallestn for which all (α1,..., α n ) are in CM(k)and eithera k ≤b 1 orb k ≤a 1, equalsk(k−1)+2, and thath(k), the smallestn for which all (α1,..., α n ) are in CM(k)and eithera 1≤b 1≤...≤a k ≤b k orb 1≤a 1≤...≤b k ≤a k , equals 2(k−1)2+2. The results forf andg rely on new theorems for regular patterns in (0, 1)-matrices that are of interest in their own right. An example is: Every upper-triangulark 2×k 2 (0, 1)-matrix has eitherk 1's in consecutive columns, each below its predecessor, ork 0's in consecutive rows, each to the right of its predecessor, and the same conclusion is false whenk 2 is replaced byk 2−1.