Turán Densities for Small Hypercubes

How small can a set of vertices in the $n$-dimensional hypercube $Q_n$ be if it meets every copy of $Q_d$? The asymptotic density of such a set (for $d$ fixed and $n$ large) is denoted by ${\gamma }_d$. It is easy to see that ${\gamma }_d\le 1\mathrm{/}(d+1)$, and it is known that ${\gamma }_d=1\mathrm{/}(d+1)$ for $d\le 2$, but it was recently shown that ${\gamma }_d<1\mathrm{/}(d+1)$ for $d\ge 8$. In this paper we show that the latter phenomenon also holds for $d=7$ and $d=6$.