= Finite linear space
A finite <linear space (geometry)> 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 gives
$$
\sum_{L\in\mathcal L}\binom{|L|}{2}=\binom{|P|}{2}.
$$
<Near-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 $|\mathcal L|\geq|P|$, with equality only for a <near-pencil> or a <finite projective plane>.<On a combinatorial problem by Nicolaas Govert de Bruijn and Paul Erdős>[Theorem1]{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>.
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