This paper establishes a connection between a certain class of Ramsey numbers for graphs and the class of symmetric block designs admitting a polarity. The main case considered here relates the projective planes over Galois fields to the Ramsey numbers
R
(
C
4
,
K
1
,
n
)
=
f
(
n
)
R({C_4},{K_{1,n}}) = f(n)
. It is shown that, for every prime power
q
,
f
(
q
2
+
1
)
=
q
2
+
q
+
2
q,f({q^2} + 1) = {q^2} + q + 2
, and
f
(
q
2
)
=
q
2
+
q
+
1
f({q^2}) = {q^2} + q + 1
.