MAXIMAL PARTIAL SPREADS AND THE MODULAR N-QUEEN PROBLEM

We prove that for any integer n in the interval (5q2 + 4q - 1)/8 less- than-or-equal-to n less-than-or-equal-to q2 - q + 2 there is a maximal partial spread of size n in PG (3, q) where q is odd and q greater-th an-or-equal-to 7. We also prove that there are maximal partial spreads of size (q2 + 3)/2 when gcd(q + 1,24)=2 or 4 and of size (q2 + 2 + 5) /2 when gcd (q + 1, 24) = 4.