Unital

Remarks

Equivalently, a unital of order $n$ is a Steiner system $S(2, n + 1, n^3 + 1)$.

Let $\Pi$ be a projective plane of order $n^2$. A unital is embedded in $\Pi$ if $X$ is a set of points of $\Pi$ and the blocks are the intersections of $X$ with lines that have size $n + 1$. In that case every line of $\Pi$ meets $X$ in $1$ or $n + 1$ points.

Example: let $q$ be a prime power and let $\F_{q^2}$ be the finite field with $q^2$ elements. In $\mathrm{PG}(2, q^2)$, the Hermitian curve $x^{q+1} + y^{q+1} + z^{q+1} = 0$, the set of isotropic points of a non-degenerate Hermitian form, has $q^3 + 1$ points. Each line meets it in $1$ or $q + 1$ points, so with the secant lines as blocks it is a unital of order $q$.

Non-example: in the Fano plane, blocks have size $3$ and any two points lie in exactly one block. But it has $7$ points rather than $2^3 + 1 = 9$, so clause 1 fails.