An intersection problem whose extremum is the finite projective space

Suppose that $\text{A}$ is a finite set-system of N elements with the property |A ∩ A′| = 0, 1 or k for any two different A, A′ ?A. We show that for N > k¹⁴

$\text{|a|=?}\text{N(N?1)(N?k)}\text{(k}{}^2\text{?k+1)(k}{}^2\text{?2k+1)}\text{+}\text{N(N?1)}\text{k(k?1)}\text{+N+1}$

where equality holds if and only if k = q + 1 (q is a prime power) $\text{N =}\text{(q}{}^{\mathrm{t+1}}\text{? 1)}\text{(q ? 1)}$ and $\text{A}$ is the set of subspaces of dimension at most two of the t-dimensional finite projective space of order q.