 
Summary: Order4(1987), 155164.
0 1987 by D. Reidel Publishing Company
155
Regressionsand Monotone Chains II:
The Poset of Integer Intervals
NOGA ALON
Tel Aviv University, Tel Avw, Israel
W. T. TROTTER
Arizona State University. Tempe, .4Z 85287, (1,S.A
and
DOUGLAS B. WEST
University ofIllinois, Urbana. IL 61801, U.S.A.
Commumcated by P. Hell
(Received: 24 September 1986; accepted: 4 March 1987)
Abstract. A regressive function (also called a regression or contractwe mapping) on a partial order
P is a function (Tmapping P to itself such that o(x) < x. A monotone kchain for CJis a kchain on
which u is orderpreserving; i.e., a chain x1 < ...
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+ lchain, 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
