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 a finite set $P$ of points and a set family $\mathcal{L}$ of subsets called lines, each containing at least two points, such that each pair of distinct points lies on exactly one line. Here the nontrivial convention also excludes a line equal to all of $P$. Two distinct lines therefore meet in at most one point, and counting pairs givesNear-pencils and finite projective planes, including the Fano plane, are examples. The De Bruijn--Erdos pair-covering inequality concerns the corresponding point degrees. The original line-count theorem proves $|\mathcal{L}|\ge |P|$, with equality only for a near-pencil or a finite projective plane.^[ref]

An exactly one-intersecting family with empty total intersection gives such a structure by using its members as points. Each original ground element supplies the set of members containing it. Discard the resulting sets of size zero or one; the others are lines, and every pair of member-points belongs to its unique shared-element line. Empty total intersection excludes an all-point line.

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.

This is a particular finite linear space configuration, rather than a synonym for every pencil of lines.

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.