criterion for a near-linear space being a linear space

Suppose 𝒮 is near-linear space with v points and b lines, and si is the number of points in the ith line, for i=1,,b. Then


and equality holds if and only if 𝒮 is a linear space.


Let N be the number of ordered pairs of points that are joined by a line. Clearly N can be no more than v(v-1), and N=v(v-1) if and only if every pair of points are joined by a line. Since two points in a near-linear space are on at most one line, we can label each pair by the line to which the two points belong to. We thus have a partitionMathworldPlanetmathPlanetmath of the N pairs into b groups, and each group is associated with a distinct line. The group corresponding to the line consisting of si points contributes si(si-1) to the total sum. Therefore

Title criterion for a near-linear space being a linear space
Canonical name CriterionForANearlinearSpaceBeingALinearSpace
Date of creation 2013-03-22 14:32:47
Last modified on 2013-03-22 14:32:47
Owner kshum (5987)
Last modified by kshum (5987)
Numerical id 9
Author kshum (5987)
Entry type Theorem
Classification msc 05B25