Bounds on Borsuk numbers in distance graphs of a special type

In 1933, Borsuk stated a conjecture, which has become classical, that the minimum number of parts of smaller diameter into which an arbitrary set of diameter $1$ in $\mathbb{R}^n$ can be partitioned is $n+1$. In 1993, this conjecture was disproved using sets of points with coordinates $0$ and $1$. Later, the second author obtained stronger counterexamples based on families of points with coordinates $-1$, $0$, and $1$. We establish new lower bounds for Borsuk numbers in families of this type.