 
Summary: Acta Math. Hung;
57 (12) (1991),6164~
A NOTE ON EUCLIDEAN RAMSEY THEORY
AND A CONSTRUCTION OF BOURGAIN
N. ALON and Y. PERES (TelAviv)
1. Qualitative facts
Let v be a fixed unit vector in a Hilbert space f2. Denote
~ = {coc~l
for a real 0
is finite. Thus, Ramsey's theorem implies
FACT 1. From any infinite sequence {co,,};=1in g2~an infinite subsequence can be
extracted, with no two vectors orthogonaI.
We will be interested in the "size" of the subsequence which can be extracted,
especially when a further restriction is put on the sequence {co,}. In particular, we
show that a subsequence of positive density cannot always be extracted.
DEFINITIONS. I. A sequence of vectors {co.} in a Hilbert space is stationary if
=(col, cot> for all i,j, n.
II. A set of integers HeN is a Van tier Corput set if every probability measure
# on the circle satisfying ~(h)=fe't@(t)=O for every hEH satisfies #{0}=0.
III. A set of integers HeN is a Poincare set if for every set ScN of positive
