criterion for a near-linear space being a linear space
Suppose is near-linear space with points and lines, and is the number of points in the th line, for . Then
and equality holds if and only if is a linear space.
Let be the number of ordered pairs of points that are joined by a line. Clearly can be no more than , and 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 partition of the pairs into groups, and each group is associated with a distinct line. The group corresponding to the line consisting of points contributes to the total sum. Therefore
|Title||criterion for a near-linear space being a linear space|
|Date of creation||2013-03-22 14:32:47|
|Last modified on||2013-03-22 14:32:47|
|Last modified by||kshum (5987)|