Is Steiner Triple System Always Regular?

Question

I know that there exists $S(2,3,n)$, when $n\equiv 1,3\mod 6$. $S(2,3,7)$ is the Fano plane and $S(2,3,9)$ is an affine plane. These two examples are in fact both regular hypergraphs, i.e. every vertices has the same degree. Are Steiner Triple Systems always regular?

Answer 1

Yes, a Steiner Triple System is always regular. Consider a STS on $n$ vertices and choose any vertex $v$. We know $v$ shares exactly one edge with every other vertex, and each edge containing $v$ contains exactly 2 other vertices. For this reason, the degree of $v$ is $\frac{n-1}{2}$. All vertices will have this same degree, so the STS is regular.