Universality of uniform Eberlein compacta

We prove that if $ \mu^+ 2^{\aleph_0}$, then there is no family of less than $ \mu^{\aleph_0}$ c-algebras of size $ \lambda$ which are jointly universal for c-algebras of size $ \lambda$. On the other hand, it is consistent to have a cardinal $ \lambda\ge \aleph_1$ as large as desired and satisfying $ \lambda^{\lambda^{++}$, while there are $ \lambda^{++}$ c-algebras of size $ \lambda^+$ that are jointly universal for c-algebras of size $ \lambda^+$. Consequently, by the known results of M. Bell, it is consistent that there is $ \lambda$ as in the last statement and $ \lambda^{++}$ uniform Eberlein compacta of weight $ \lambda^+$ such that at least one among them maps onto any Eberlein compact of weight $ \lambda^+$ (we call such a family universal). The only positive universality results for Eberlein compacta known previously required the relevant instance of $ GCH$ to hold. These results complete the answer to a question of Y. Benyamini, M. E. Rudin and M. Wage from 1977 who asked if there always was a universal uniform Eberlein compact of a given weight.