  A $k$-independent set in a connected graph is a set of vertices such that any
two vertices in the set are at distance greater than $k$ in the graph. The
$k$-independence number of a graph, denoted $\alpha_k$, is the size of a
largest $k$-independent set in the graph. Recent results have made use of
polynomials that depend on the spectrum of the graph to bound the
$k$-independence number. They are optimized for the cases $k=1,2$. There are
polynomials that give good (and sometimes) optimal results for general $k$,
including case $k=3$. In this paper, we provide the best possible bound that
can be obtained by choosing a polynomial for case $k=3$ and apply this bound to
well-known families of graphs including the Hamming graph.
