  Fulton's conjecture for the moduli space of stable pointed rational curves,
\bar{M}_{0,n}, claims that a divisor non-negatively intersecting all F-curves
is linearly equivalent to an effective sum of boundary divisors. Our main
result is a proof of Fulton's conjecture for n=7. A key ingredient in the proof
is an \binom{n}{4} dimensional-subspace of the Neron-Severi space of
\bar{M}_{0,n}, defined by averages of Keel relations, for which we prove
Fulton's conjecture for all n.
