By definition, a Steiner Triple System (STS) is a $t-(v,k,\lambda)$ design with blocks of size 3 ($k=3$) such that each pair of points meets in exactly 1 block ($t=2$ and $\lambda=1$). Let us denote the set of points of an STS with $V$ and the set of blocks with $\mathcal{B}$. A complete arc in an STS is a subset of $V$, say $S\subset V$, such that $S$ is both:
- independent, i.e., no block $B\in\mathcal{B}$ is contained in $S$.
- and spanning, i.e., every point $x\in V \setminus S$ is spanned by a pair $\{s,t\}$ in $S$ which means that $\{s,t,x\}\in\mathcal{B}$.
With that said, there are multiple papers, e.g., [Sauer & Schonheim, 1969] that characterize the maximal size of a complete arc $|S|$ in an STS of $v$ points. This maximal size denoted by $T(v)$ is
$$T(v)= \left\{ \begin{array}{ll} (v+1)/2, & \text{if } v \equiv 3,7\ (\text{mod}\ 12) \\ (v-1)/2, & \text{if } v \equiv 1,9\ (\text{mod}\ 12) \end{array} \right. $$
I am looking for a systematic construction of STS that has a complete arc with the maximal size, $T(v)$. The paper [Sauer & Schonheim, 1969] proposes this method
Partition the set $V$ into subsets $S$ and $V\setminus S$, with $|S|=T(v)$ (either $(v+1)/2$ or $(v-1)/2$, depending on the case). Then, construct triples which contain the pairs of $S$ by assigning each pair of $S$ to a certain element of $V\setminus S$.
Unfortunately, this method is not complete, i.e., there are missing details on how to construct the STS. For example, consider $v=7$ points in $V=\{1,2,3,4,5,6,7\}$. Let us pick $S=\{1,2,3,4\}$. Creating all pairs of points in $S$, we have: $$\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\}$$ Then, I completed each of these sets with a point from $V\setminus S = \{5,6,7\}$ such that each pair of points meet in no more than $\lambda=1$ blocks as: $$\{1,2,5\},\{1,3,6\},\{1,4,7\},\{2,3,7\},\{2,4,6\},\{3,4,5\}$$ Clearly, this is not an STS since the pairs $\{5,6\},\{5,7\}$ and $\{6,7\}$ do not meet in any block. So, I had to add the block $\{5,6,7\}$ and then the following blocks $$\mathcal{B}=\{\{1,2,5\},\{1,3,6\},\{1,4,7\},\{2,3,7\},\{2,4,6\},\{3,4,5\},\{5,6,7\}\}$$ characterize a valid STS of $v=7$ points.
This is an easy example as there are only $7$ points. But, what is a systematic method to construct such an STS? Can you maybe point to some references that discuss this construction in detail?