  Let $G = (V,E)$ be a graph and $k \ge 0$ an integer. A $k$-independent set $S
\subseteq V$ is a set of vertices such that the maximum degree in the graph
induced by $S$ is at most $k$. With $\alpha_k(G)$ we denote the maximum
cardinality of a $k$-independent set of $G$. We prove that, for a graph $G$ on
$n$ vertices and average degree $d$, $\alpha_k(G) \ge \frac{k+1}{\lceil d
\rceil + k + 1} n$, improving the hitherto best general lower bound due to Caro
and Tuza [Improved lower bounds on k-independence, J. Graph Theory 15 (1991),
99-107].
