Revisiting $\mathfrak{b}$ and $\mathfrak{d}$ through Interval Structures

We investigate a family of relational systems arising from interval partitions of $\omega$, inspired by Vojtáš’s characterization of the bounding and dominating numbers. By varying the underlying asymptotic quantifiers and interval constraints, we obtain several natural interval-type generalizations. We show that the universal variants are remarkably robust: in all the discrete, colored, restricted, bounded, and measure-theoretic settings considered here, the associated bounding and dominating numbers coincide with the classical invariants $\mathfrak{b}$ and $\mathfrak{d}$. In contrast, the existential variants systematically reverse these invariants, yielding that the bounding number coincides with $\mathfrak{d}$ and the dominating number coincides with $\mathfrak{b}$.