Codegree problems for projective geometries

The codegree density γ ( F ) of an r-graph F is the largest number γ such that there are F-free r-graphs G on n vertices such that every set of r − 1 vertices is contained in at least ( γ − o ( 1 ) ) n edges. When F = PG 2 ( 2 ) is the Fano plane Mubayi showed that γ ( F ) = 1 / 2 . This paper studies γ ( PG m ( q ) ) for further values of m and q. In particular we have an upper bound γ ( PG m ( q ) ) ⩽ 1 − 1 / m for any projective geometry. We show that equality holds whenever m = 2 and q is odd, and whenever m = 3 and q is 2 or 3. We also give examples of 3-graphs with codegree densities equal to 1 − 1 / k for all k ⩾ 1 .