  We consider (near-)critical percolation on the square lattice. Let M_n be the
size of the largest open cluster contained in the box [-n,n]^2, and let pi(n)
be the probability that there is an open path from O to the boundary of the
box. It is well-known that for all 0< a < b the probability that M_n is smaller
than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are
bounded away from 0 as n tends to infinity. It is a natural question, which
arises for instance in the study of so-called frozen-percolation processes, if
a similar result holds for the probability that M_n is between an^2 pi(n) and
bn^2 pi(n). By a suitable partition of the box, and a careful construction
involving the building blocks, we show that the answer to this question is
affirmative. The `sublinearity' of 1/pi(n) appears to be essential for the
argument.
