First distribution invariants and EKR theorems

Abstract

It is shown by a simple counting argument that, in a projective space P_n−1, any set of [(n−1)/(d−1)] distinct (d−1)-subspaces of P_n−1, d|n, contains a d-spread. A weight function on P_n−1, is a real-valued function on the set In of points of P_n−1 such that the sum of the values on all points of I_n is nonnegative. The weight of any subset of I_n is the sum of the weights of all the points in it. It is shown that the number of (d−1)-subspaces in P_n−1 with nonnegative weights is at least [(n−1)/(d−1)]. The case of equality is characterized by using the Erdös-Ko-Rado theorem. These results are then applied to study the first distribution invariant of J_q(n, d). Analogues of these results are proved for sets and affine spaces when n≥2d. In the case of affine spaces the problem is essentially solved.