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A linear space is a structure in incidence geometry with a set of points and a collection of subsets called lines: any two distinct points lie on exactly one line, and every line contains at least two points. The lines here are incidence-theoretic objects, without a required coordinate realization. A finite linear space imposes finiteness; its nontrivial form excludes a line containing all points. This is distinct from a vector space, whose structure involves addition and scalar multiplication.

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.

Discard ground points lying in no set. For every point $x$ that lies in at least two members, formEvery pair of indices lies in exactly one $L_x$, and no $L_x$ equals $[n]$ because the total intersection is empty. Thus the $L_x$ form a finite linear space on the $n$ indices. The number of its lines through index $i$ is at most $a_i$, with equality unless $A_i$ contains private points.

The De Bruijn--Erdos pair-covering inequality says that if $r_i$ is the number of lines through point $i$ in a nontrivial finite linear space on $n$ points, thenApplying it here givesMoreover, a pair of ground points can lie in at most one $A_i$, since two different members meet in only one point. HenceCombining the inequalities gives $\left(\genfrac{}{}{0ex}{}{m}{2}\right)\ge \left(\genfrac{}{}{0ex}{}{n}{2}\right)$, and therefore $m\ge n$.