 
Summary: JOURNAL OF COMBINATORIAL THEORY, Series A 52, 275287 (1989)
Ascending Waves
N. ALON*
Bell Communication Research, Morrisiown, New Jersey 07960 and
Deparrment of Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel
AND
JOEL SPENCER
Department of Mathematics, SUNY at Stony Brook,
Stony Brook, New York 1I794
Communicated by the Managing Editors
Received September 15, 1987
A sequence of integers x,
if x~+~x~
integer such that any 2coloring of {1,2, .... f(k)} contains a monochromatic
ascending wave of length k. Settling a problem of Brown, Erdiis, and Freedman we
show that there are two positive constants c,, c2 such that c,k3
all k > 1. Let g(n) be the largest integer k such that any set A c { 1,2, .... n} of car
dinality IA( an/2 contains an ascending wave of length k. We show that there are
two positive constants cj and c4 such that c,(log n)2/log log n < g(n) < c,(log n)2 for
all n 2 1. 0 1989 Academic Press, Inc.
