Linearity of free resolutions of monomial ideals

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