  It was previously known, by work of Smith-Swanson and of Sharp-Nossem, that
the linear growth property of primary decompositions of Frobenius powers of
ideals in rings of prime characteristic has strong connections to the
localization problem in tight closure theory. The localization problem has
recently been settled in the negative by Brenner and Monsky, but the linear
growth question is still open. We study growth of primary decompositions of
Frobenius powers of dimension one homogeneous ideals in graded rings over
fields. If the ring is positively graded we prove that the linear growth
property holds. For non-negatively graded rings we are able to show that there
is a "polynomial growth". We present explicit primary decompositions of
Frobenius powers of an ideal, which were known to have infinitely many
associated primes, having this linear growth property. We also discuss some
other interesting examples.
