  We study monomial ideals with linear presentation or partially linear
resolution. We give combinatorial characterizations of linear presentation for
square-free ideals of degree 3, and for primary ideals whose resolutions are
linear except for the last step (the "almost linear" case). We also give sharp
bounds on Castelnuovo-Mumford regularity and numbers of generators in some
cases.
  It is a basic observation that linearity properties are inherited by the
restriction of an ideal to a subset of variables, and we study when the
converse holds. We construct fractal examples of almost linear primary ideals
with relatively few generators related to the Sierpi\'nski triangle. Our
results also lead to classes of highly connected simplicial complexes $\Delta$
that can not be extended to the complete $\dim \Delta$-skeleton of the simplex
on the same variables by shelling.
