Full upper saturation spectrum for Berge stars with l ≥ 6

Determine the full upper range of the 3-uniform saturation spectrum for Berge-K_{1,l} for every integer l ≥ 6, overcoming the limitations of the casework based on the edge difference between the l-lantern L_l and three copies of K^{(3)}_l.

Background

The paper completely determines the 3-uniform saturation spectrum of Berge-K_{1,5} when 5 divides n, but for Berge-K_{1,l} with l ≥ 6 it establishes only that all but a constant number of values in the spectrum occur. The unresolved portion is the upper range near the extremal number.

The authors explain that the l = 5 argument is tractable because replacing three copies of the complete 3-graph K^{(3)}_5 with the 5-lantern L_5 changes the edge count by only seven, permitting a finite seven-case analysis. For larger l, this edge difference grows, making the analogous casework infeasible; the problem is therefore to develop methods capable of determining the entire upper range for all l ≥ 6.