Let
α
→
(
β
)
m
r
\alpha \rightarrow (\beta )_m^r
denote the property: if
A
A
is an
α
\alpha
–large set of natural numbers and
[
A
]
r
[A]^r
is partitioned into
m
m
parts, then there exists a
β
\beta
–large subset of
A
A
which is homogeneous for this partition. Here the notion of largeness is in the sense of the so–called Hardy hierarchy. We give a lower bound for
α
\alpha
in terms of
β
,
m
,
r
\beta ,m,r
for some specific
β
\beta
.