Let
X
X
be a compact Hausdorff space and
κ
\kappa
its Souslin number.
2
^{2}
We prove that if
α
\alpha
is a cardinal such that either
α
\alpha
and
cf
(
α
)
\operatorname {cf} (\alpha )
are greater than
κ
\kappa
and strongly
κ
\kappa
-inaccessible or else
α
\alpha
is regular and greater than
κ
\kappa
, then
X
X
has
(
α
,
α
κ
⏝
)
(\alpha , \sqrt [\underparen {\kappa }]{\alpha })
caliber. Restricting our interest to the category of compact spaces
X
X
with
S
(
X
)
=
ω
+
S(X) = {\omega ^ + }
(i.e.
X
X
satisfy the countable chain condition), the above statement takes, under G.C.H., the following form. For any compact space
X
X
with
S
(
X
)
=
ω
+
S(X) = {\omega ^ + }
, we have that (a) if
α
\alpha
is a cardinal and
cf
(
α
)
\operatorname {cf} (\alpha )
does not have the form
β
+
{\beta ^ + }
with
cf
(
β
)
=
ω
\operatorname {cf} (\beta ) = \omega
, then
α
\alpha
is caliber for the space
X
X
. (b) If
ε
=
β
+
\varepsilon = {\beta ^ + }
and
cf
(
β
)
=
ω
\operatorname {cf} (\beta ) = \omega
then
(
α
,
β
)
(\alpha ,\,\beta )
is caliber for
X
X
. A related example shows that the result of (b) is in a sense the best possible.