Exactly one-intersecting family

An exactly one-intersecting family satisfies $|A\cap B|=1$ for every two distinct members. It is nontrivial when the intersection of all its members is empty.

A finite linear space consists of finitely many points and proper subsets called lines such that every two points lie on exactly one line. The sets of family indices containing each ground point turn a nontrivial exactly one-intersecting family into a finite linear space.

If $r_x$ denotes the number of lines through $x$ in a nontrivial finite linear space on $v$ points, thenTo prove it, let $k_L$ be the line sizes. The inequalities $k_L\le r_x$ whenever $x\notin L$, counted at each integer threshold, givefor every $t\ge 1$. Summing in $t$ and using $\left(\genfrac{}{}{0ex}{}{s}{2}\right)={\sum }_{t\ge 1}(s-t{)}_{+}$ giveswhere the last identity counts pairs of points by their unique line.

A near-pencil on $v$ points has one line containing $v-1$ points and $v-1$ two-point lines joining the remaining point to each point of the large line.

A finite projective plane of order $q$ has $q^2+q+1$ points and the same number of lines. Every line contains $q+1$ points, every point lies on $q+1$ lines, and every two points or two lines determine a unique line or intersection point respectively.