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 = PG2(2) is the Fano plane Mubayi showed that γ(F ) = 1/2. This paper studies γ(PGm(q)) for further values of m and q. In particular we have an upper bound γ(PGm(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.