Regressions and Monotone Chains II: The Poset of Integer Intervals

A regressive function (also called a regression or contractwe mapping) on a partial order P is a function (T mapping P to itself such that o(x) < x. A monotone k-chain for CJ is a k-chain on which u is order-preserving; i.e., a chain x1 < ... <XL such that a(.~, ) d ... < a(xk). Let P,be the poset of integer intervals {i, i + 1, , m} contained in { 1,2, . . . . n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P, has a monotone k+ l-chain, let t(x, j) be defined by t(x, O)= 1 and t(x,J)=X’(‘x’-I’. Then f(k) exists for all k (originally proved by D. White), and t(2,k)<f(k)<t(e+ek,k), where &k-+0 as k+m. Alternatively, the largest k such that every regression on P, is guaranteed to have a monotone k-chain lies between lg*(n) and lg*(n)2, inclusive, where lg*(n) is the number of applications of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element. AMS subject classifications (1980). 05(355,06AlO, 625.