Abstract. Let n and t be positive integers with t < n, and let q be a prime power. A partial (t − 1)-spread of PG(n − 1, q) is a set of (t − 1)-dimensional subspaces of PG(n − 1, q) that are pairwise disjoint. Let r ≡ n (mod t) with 0 ≤ r < t, and let Θi = (q i − 1)/(q − 1). We essentially prove that if 2 ≤ r < t ≤ Θr, then the maximum size of a partial (t − 1)-spread of PG(n − 1, q) is bounded from above by (Θn − Θt+r)/Θt + q r − (q − 1)(t − 3) + 1. We actually give tighter bounds when certain divisibility conditions are satisfied. These bounds improve on the previously known upper bound for the maximum size partial (t − 1)-spreads of PG(n − 1, q); for instance, when ⌈ Θr 2 ⌉ + 4 ≤ t ≤ Θr and q > 2. The exact value of the maximum size partial (t − 1)-spread has been recently determined for t > Θr by the authors of this paper (see ).

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