Dividing A Set Of Lines
Find the smallest positive integer \(N\) that has the following property.
Suppose we have a set of \(2014\) lines in 3 dimensions that pass through the origin, and no plane contains more than three lines. Then we can divide the lines into \(N\) groups, so that every line is perpendicular to at most one line in its group.
Details and assumptions
The statement has to be true for any set of \(2014\) lines, and not just a particular set that you chose.