Binomial partial Steiner triple systems with complete graphs: structural problems

In the paper we study the structure of hyperplanes of so called binomial partial Steiner triple systems (BSTS’s, in short) i.e. of configurations with $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ points and $\left(\genfrac{}{}{0px}{}{n}{3}\right)$ lines, each line of the size $3$. Consequently, a BSTS has $n-2$ lines through each of its points.