Partial covers of

In this paper, we show that a set of q + a hyperplanes, q > 13 , a ≤ ( q − 10 ) / 4 , that does not cover PG ( n , q ) , does not cover at least q n − 1 − a q n − 2 points, and show that this lower bound is sharp. If the number of non-covered points is at most q n − 1 , then we show that all non-covered points are contained in one hyperplane. Finally, using a recent result of Blokhuis, Brouwer and Szőnyi [8], we remark that the bound on a for which these results are valid can be improved to a < ( q − 2 ) / 3 and that this upper bound on a is sharp.