Partial covers of PG(n,q)

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 qn−1−aqn−2 points, and show that this lower bound is sharp. If the number of non-covered points is at most qn−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.