Definitions: Flock, Linear Flock, Thick Cone, Carrier Plane, Baseline


Theorem 9.1: If C(V,S) is a thick cone, then any flock of it which contains more than q -[q+2/2] planes that meet in a common line is a linear flock.

Pf: Let F be a flock of a thick cone C(V,S). Suppose that x of the planes of F meet at a common line l, with 2x < q. Let be a plane of F which does not contain l. The line l meets at a point P. Each of the q+1 planes through l meet in a line passing through P. Let S' be the intersection of with C(V,S).

Now, x of the lines through P are the intersections of with planes of the flock. No point of S' can lie on these x lines. One of the other lines through P is the intersection of with the plane determined by l and V. No point of S' can lie on this line either, for if there was such a point there would be a point of S (the projection of this point from V into the carrier plane) on the baseline which is the projection of l into the carrier plane. So, the points of S' all lie on q - x lines through P. By projecting from V into the carrier plane, we see that the points of S all lie on q - x concurrent lines. Since the cone is thick we have:

q - x [q+2/2].
Thus, xq - [q+2/2]. So, if x > q -[q+2/2] we must have x = q, i.e., F is a linear flock.


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