The Turلn problem for projective geometries

We consider the following Turán problem. How many edges can there be in a ( q + 1 ) -uniform hypergraph on n vertices that does not contain a copy of the projective geometry PG m ( q ) ? The case q = m = 2 (the Fano plane) was recently solved independently and simultaneously by Keevash and Sudakov (The Turán number of the Fano plane, Combinatorica, to appear) and Füredi and Simonovits (Triple systems not containing a Fano configuration, Combin. Probab. Comput., to appear). Here we obtain estimates for general q and m via the de Caen–Füredi method of links combined with the orbit-stabiliser theorem from elementary group theory. In particular, we improve the known upper and lower bounds in the case q = 2 , m = 3 .