Characterize the saturation spectrum of 3-uniform Berge triangles

Characterize the 3-uniform saturation spectrum of Berge-K_3, and determine whether it contains all but finitely many values near the top of the range.

Background

The paper studies saturation spectra for 3-uniform Berge stars. It notes that other hypergraphs are natural subjects for investigation but difficult to analyze because exact extremal and saturation numbers are known for very few forbidden hypergraphs.

The authors identify 3-uniform Berge-K_3 as a possible example because prior work establishes exact formulas for both its saturation number and extremal number. They conjecturally suspect that, analogously to the star case, only finitely many values near the extremal end may be absent from its saturation spectrum.