If
G
G
and
H
H
are graphs, define the Ramsey number
r
(
G
,
H
)
r(G,H)
to be the least number
p
p
such that if the edges of the complete graph
K
p
{K_p}
are colored red and blue (say), either the red graph contains
G
G
as a subgraph or the blue graph contains
H
H
. Let
m
G
mG
denote the union of
m
m
disjoint copies of
G
G
. The following result is proved: Let
G
G
and
H
H
have
k
k
and
l
l
points respectively and have point independence numbers of
i
i
and
j
j
respectively. Then
N
−
1
⩽
r
(
m
G
,
n
H
)
⩽
N
+
C
N - 1 \leqslant r(mG,nH) \leqslant N + C
, where
N
=
k
m
+
l
n
−
m
i
n
(
m
i
,
m
j
)
N = km + ln - min(mi,mj)
and where
C
C
is an effectively computable function of
G
G
and
H
H
. The method used permits exact evaluation of
r
(
m
G
,
n
H
)
r(mG,nH)
for various choices of
G
G
and
H
H
, especially when
m
=
n
m = n
or
G
=
H
G = H
. In particular,
r
(
m
K
3
,
n
K
3
)
=
3
m
+
2
n
r(m{K_3},n{K_3}) = 3m + 2n
when
m
⩾
n
,
m
⩾
2
m \geqslant n,m \geqslant 2
.