Copies of $c_0(\tau)$ in Saphar tensor products

Let $X, Y$ be Banach spaces, τ an infinite cardinal and $1 \leq p < \infty $. We extend a result by E. Oja by showing that if $X$ has a boundedly complete unconditional basis and either $X \widehat{\otimes}_{g_p} Y$ or $X \widehat{\otimes}_{\varepsilon _p} Y$ contains a complemented copy of $c_0(\tau )$, then $Y$ contains a complemented copy of $c_0(\tau )$. We show also that if α is a uniform crossnorm, $X \widehat{\otimes}_\alpha Y$ contains a (complemented) copy of $c_0(\tau )$ and the cofinality of τ is strictly greater than the density of $X$, then $Y$ also contains a (complemented) copy of $c_0(\tau )$. As an application, we obtain a result concerning complemented copies of $\ell _1(\tau )$ in $X \widehat{\otimes}_\alpha Y$.