Minimal vertices for PR complexes of a given degree type

Determine, for each sequence d of positive integers, the minimum number n such that there exists a simplicial complex Δ whose dual Stanley-Reisner ideal I_{Δ*} has a pure resolution with degree type d and Δ has exactly n vertices. Equivalently, ascertain the smallest initial shift c0(d) for pure Betti diagrams of dual Stanley-Reisner ideals with degree type d.

Background

Pure-resolution (PR) complexes are simplicial complexes whose dual Stanley-Reisner ideals have pure minimal free graded resolutions. The degree type records the differences between consecutive shifts in such a resolution. Earlier in the thesis (Question 5.1), the author asks for the lowest n such that the cone D_n contains a pure diagram of degree type d; Section 7.1 clarifies that this remains unresolved in most cases.

The thesis provides algorithms that yield PR complexes for any prescribed degree type, thus establishing existence and upper bounds on the number of vertices, and introduces several families (cycle, intersection, partition complexes) that offer improved bounds for special degree types. However, finding the exact minimal n(d) is open in general.