Determine an explicit formula for the minimum-degree threshold function

Determine an explicit formula for the function α(d), defined for each nonnegative integer d as the largest real number such that every finite abstract simplicial complex S satisfying |S| ≤ α(d)|V(S)| has a vertex of degree at most d.

Background

For a finite abstract simplicial complex S, α(d) measures the threshold ratio between the number of faces and vertices that forces the existence of a vertex of degree at most d. Equivalently, a complex whose minimum vertex degree exceeds d must have more than α(d)|V(S)| faces.

The paper determines α(2d−m) for all integers d ≥ m ≥ 1 and also establishes the previously conjectured value α(11). However, the authors explicitly state that no formula for α(d) valid for all nonnegative integers d is currently available, leaving the general determination of this function unresolved.