Summary: \textit{D. Karger, R. Motwani} and \textit{M. Sudan} [J. ACM 45, 246--265 (1998; Zbl 0904.68116)] introduced the notion of a vector coloring of a graph. In particular, they showed that every \(k\)-colorable graph is also vector \(k\)-colorable, and that for constant \(k\), graphs that are vector \(k\)-colorable can be colored by roughly \(\Delta^{1- 2/k}\) colors. Here \(\Delta\) is the maximum degree in the graph and is assumed to be of the order of \(n^{\delta}\) for some \(0 < \delta < 1\). Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer \(k\) there are graphs that are vector \(k\)-colorable but do not have independent sets significantly larger than \(n/\Delta^{1- 2/k}\) (and hence cannot be colored with significantly fewer than \(\Delta^{1- 2/k}\) colors). For \(k = O(\log n/\log\log n)\) we show vector \(k\)-colorable graphs that do not have independent sets of size \((\log n)^{c}\), for some constant \(c\). This shows that the vector chromatic number does not approximate the chromatic number within factors better than \(n/\operatorname{polylog} n\). As part of our proof, we analyze ``property testing'' algorithms that distinguish between graphs that have an independent set of size \(n/k\), and graphs that are ``far'' from having such an independent set. Our bounds on the sample size improve previous bounds of \textit{O. Goldreich, S. Goldwasser} and \textit{D. Ron} [J. ACM 45, 653--750 (1998; Zbl 1065.68575)] for this problem.