  We prove the following three closely related results. The first is that every
finite simple group has a profinite presentation with 2 generators and at most
18 relations. The second is that if G is a finite simple group, F a field and M
an FG-module, then the dimension of the second cohomology group of G with
coefficients in M is at most 17.5 times the dimension of M. The third result is
that we may replace 17.5 by 18.5 as long as M is faithful irreducible G-module.
These last two results answer conjectures of Holt.
