  A relatively compressed algebra with given socle degrees is an Artinian
quotient $A$ of a given graded algebra $R/\fc$, whose Hilbert function is
maximal among such quotients with the given socle degrees. For us $\fc$ is
usually a ``general'' complete intersection and we usually require that $A$ be
level. The precise value of the Hilbert function of a relatively compressed
algebra is open, and we show that finding this value is equivalent to the
Fr\"oberg Conjecture. We then turn to the minimal free resolution of a level
algebra relatively compressed with respect to a general complete intersection.
When the algebra is Gorenstein of even socle degree we give the precise
resolution. When it is of odd socle degree we give good bounds on the graded
Betti numbers. We also relate this case to the Minimal Resolution Conjecture of
Mustata for points on a projective variety. Finding the graded Betti numbers is
essentially equivalent to determining to what extent there can be redundant
summands (i.e. ``ghost terms'') in the minimal free resolution, i.e. when
copies of the same $R(-t)$ can occur in two consecutive free modules. This is
easy to arrange using Koszul syzygies; we show that it can also occur in more
surprising situations that are not Koszul. Using the equivalence to the
Fr\"oberg Conjecture, we show that in a polynomial ring where that conjecture
holds (e.g. in three variables), the possible non-Koszul ghost terms are
extremely limited. Finally, we use the connection to the Fr\"oberg Conjecture,
as well as the calculation of the minimal free resolution for relatively
compressed Gorenstein algebras, to find the minimal free resolution of general
Artinian almost complete intersections in many new cases. This greatly extends
previous work of the first two authors.
