On Bohman's conjecture related to a sum packing problem of Erdos

Motivated by a sum packing problem of Erdos, Bohman discussed an extremal geometric problem which seems to have an independent interest. Let $H$ be a hyperplane in $\mathbb R^n$ such that $H\cap\{0,\pm1\}^n=\{0^n\}$. The problem is to determine

Bohman (1996) conjectured that

We show that for some constants $c_1,c_2$ we have $c_1(2,538)^n<f(n)< c_2(2,723)^n$--disproving the conjecture. We also consider a more general question of the estimation of $\vert H\cap\{0,\pm1,\dots,\pm m\}\vert$, when $H\cap\{0,\pm1,\dots,\pm k\}=\{0^n\}$, $m>k>1$.