Trivial colors in colorings of Kneser graphs

We show that any proper coloring of a Kneser graph KG(n,k) with n-2k+2 colors contains a trivial color class (i.e., a color class consisting of sets that all contain a fixed element), provided n>(2+epsilon)k(2), where epsilon -> 0 as k ->infinity. This bound is essentially tight. This is a consequence of a more general result on the minimum number of non-trivial color classes needed to properly color KG(n,k).