New bounds for partial spreads of H(2d - 1, q2) and partial ovoids of the Ree-Tits octagon

Two results are obtained that give upper bounds on partial spreads and partial ovoids respectively. The first result is that the size of a partial spread of the Hermitian polar space H(3, q) is at most ( 2p+p 3 )t +1, where q = p, p is a prime. For fixed p this bound is in o(q), which is asymptotically better than the previous best known bound of (q + q + 2)/2. Similar bounds for partial spreads of H(2d − 1, q), d even, are given. The second result is that the size of a partial ovoid of the Ree-Tits octagon O(2) is at most 26 + 1. This bound, in particular, shows that the Ree-Tits octagon O(2) does not have an ovoid.