Cross-Monotone Subsequences

Two finite real sequences (aI , . . . . ak) and (b, , . . . . bk) are cross-monotone if each is nondecreasing and ai + 1 -ci > bi + 1 bt for all i < k. A sequence ((x, , . . . , on) of nondecreasing reals is in class CM(k) if it has disjoint k-term subsequences that are cross-monotone. The paper shows that f(k), the smallest n such that every nondecreasing (o, , . . . , on) is in CM(k), is bounded between about ka/4 and k”/2. It also shows that g(k), the smallest n for which all (CY~, . . . . on) are in CM(k) and either ak < b, or bk < a,, equals k(k 1) + 2, and that h(k), the smallest n for which all (o, , . . , orn) are in CM(k) and either a, < b, < ‘1. G ak < bk or b, < a, < ... < bk Q ak, equals 2(k 1)’ + 2. The results for f and g rely on new theorems for regular patterns in (0, 1)-matrices that are of interest in their own right. An example is: Every upper-triangular kz x ka (0, l)-matrix has either k I’s in consecutive columns, each below its predecessor, or k O’s in consecutive rows, each to the right of its predecessor, and the same conclusion is false when k’ is replaced by k’ 1. AMS (MOS) subject classifications (1980): Primary: 06A99; secondary: 05A05