Characterize posets forcing unbounded induced saturation numbers

Characterize the intrinsic properties that a finite poset must possess for a saturated family separating the ground set to exist, thereby determining which finite posets have unbounded induced saturation number.

Background

For a finite poset, the induced saturation number is known to be either bounded or subject to nontrivial lower bounds when saturated families separate the ground set. Several sufficient structural conditions for unboundedness are known, including the unique cover twin property and related constructions, but these conditions are not exhaustive; the poset 2C2 provides a counterexample to their exhaustiveness. The paper explicitly identifies the general structural characterization of posets with unbounded saturation number as unresolved.

References

They also made the following conjecture.

— The Bounded-Linear Dichotomy Holds for all 4 Point Posets  (2610.06527 - Ivan et al., 5 Oct 2026) in Conjecture 1, Section 1, Introduction

However, this tells us nothing about the intrinsic properties a poset must have in order for such a saturated family to exist. In fact, this question is very much open.

— The Bounded-Linear Dichotomy Holds for all 4 Point Posets  (2610.06527 - Ivan et al., 5 Oct 2026) in Section 1, Introduction