Theorem 1.4
(Canonical Ramsey Theorem)
.
Assuming that:
ℕ
(
2
)
is coloured (possibly with an infinite number of colours)
Then
there exists an infinite set
M
such that one of the following holds:
(i)
c
is constant on
M
.
(ii)
c
is injective on
M
.
(iii)
c
(
{
i
,
j
}
)
=
c
(
{
k
,
l
}
)
if and only if
i
=
k
for
i
<
j
,
k
<
l
in
M
.
(iv)
c
(
{
i
,
j
}
)
=
c
(
{
k
,
l
}
)
if and only if
j
=
l
, for all
i
<
j
,
k
<
l
.