New bounds for partial spreads of [formula omitted] 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 2 ) is at most ( ( 2 p 3 + p ) / 3 ) t + 1 , where q = p t , p is a prime. For fixed p this bound is in o ( q 3 ) , which is asymptotically better than the previous best known bound of ( q 3 + q + 2 ) / 2 . Similar bounds for partial spreads of H ( 2 d − 1 , q 2 ) , d even, are given. The second result is that the size of a partial ovoid of the Ree–Tits octagon O ( 2 t ) is at most 26 t + 1 . This bound, in particular, shows that the Ree–Tits octagon O ( 2 t ) does not have an ovoid. [ABSTRACT FROM AUTHOR]