Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(K

^{n})
or equivalently the n-fold injective tensor product of the C(K)s or the Banach space of vector
valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of
complemented copies of c

_{0}(ω

_{1}) in C(K

^{n}) under the hypothesis that C(K)
contains an isomorphic copy of c

_{0}(ω

_{1}).
This is related to the results of E. Saab and P. Saab that the injective tensor product of X and Y
contains a complemented copy of c

_{0}, if one of the infinite dimensional Banach spaces X or Y
contains a copy of c

_{0} and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that
if C(K) has density ω

_{1} and contains a copy of c

_{0}(ω

_{1}), then C(K×K) contains
a complemented copy c

_{0}(ω

_{1}).

The main result is that under the assumption of the club principle for every n in N there is a compact Hausdorff
space K

_{n} of weight ω

_{1} such that C(K

_{n}) is Lindelof in the weak topology,
C(K

^{n}) contains a copy of c

_{0}(ω

_{1}), C(K

_{n}^{n}) does
not contain a complemented copy of c

_{0}(ω

_{1}) while C(K

_{n}^{n+1})
does contain a complemented copy of c

_{0}(ω

_{1}). This shows that
additional set-theoretic assumptions in Galego and Hagler's nonseparable version
of Cembrano and Freniche's theorem are necessary as well as clarifies in the negative
direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach
spaces must be weakly pcc.