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A [[Steiner system]] $\mathrm{S}(2,3,n)$, that is, a set of size $n$ with a distinguished collection of subsets of size $3$ ("blocks") such that every subset of size $2$ is contained in exactly one block; denoted $\mathrm{STS}(n)$.  Such a system exists if and only if $n \equiv 1,3 \pmod 6$: this was already established by Revd T.P. Kirkman in 1846.  

The [[projective plane]] $\mathrm{P}(2,2)$ of order $2$, consisting of $7$ points and $7$ lines each containing $3$ points, in which any two points determine a unique line, is an $\mathrm{STS}(7)$.

A ''resolution'' of a design is a partition of its blocks into "parallel" classes, such that element of the underlying set is contained in just one block of each class: a ''resolvable design'' is one with a resolution.  A resolvable Steiner triple system is a '''Kirkman triple system''' $\mathrm{KTS}(n)$.  Such systems exist if and only if $n \equiv 3 \pmod 6$.  The [[Kirkman schoolgirls problem]], of finding a $\mathrm{KTS}(15)$, was one of the [[classical combinatorial problems]], solved by T.P. Kirkman in 1850.  

A Steiner triple system gives rise to a [[quasi-group]] structure on the underlying set, defined by the binary operation $x \cdot x = x$ and $x \cdot y = z$ when $x\ne y$ and $\{x,y,z\}$ is the unique block containing $\{x,y\}$. 


====References====
* Thomas Beth, Dieter Jungnickel, Hanfried Lenz, "Design theory", Cambridge University Press (1986) {{ZBL|0602.05001}}
* Anne Penfold Street, Deborah J. Street, "Combinatorics of experimental design", Clarendon Press (1987) {{ISBN|0-19-853255-5}} {{ZBL|0622.05001}}