Before revision:  Our main result is that an r-round algorithm for cardinality-constrained monotone maximization cannot achieve a factor better than 1 - 1/e - \Omega(\min \{ 1{r}, \log^2 n{r^3} \}), for any r < n^c (where c>0 is some constant).
After revision:  Our main result is that an r-round algorithm for cardinality-constrained monotone maximization cannot achieve an approximation factor better than 1 - 1/e - \Omega(\min \{ 1{r}, \log^2 n{r^3} \}), for any r < n^c (where c>0 is some constant).