Are there models of ZFC in which $\mathfrak{r}$ is strictly less than $\mathfrak{s}$? I've not been able to find any forcings that end up with this result.


Here $\mathfrak{r}$ is the reaping number $\lvert\lvert([\omega]^\omega,[\omega]^\omega,\text{does not split})\rvert\rvert$ and $\mathfrak{s}$ is the splitting number, its dual.



Can we force $\mathfrak{r}<\mathfrak{s}$?
The inequality $\mathfrak{r} \leq \mathfrak{u}$ is provable in ZFC (because every base for an ultrafilter is a reaping family). Blass and Shelah proved the consistency of $\mathfrak{u} < \mathfrak{s}$ in



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> *Blass, Andreas; Shelah, Saharon*, [**There may be simple $P\_{\aleph\_1}$- and $P\_{\aleph\_2}$-points and the Rudin-Keisler ordering may be downward directed**](https://doi.org/10.1016/0168-0072(87)90082-0), Ann. Pure Appl. Logic 33, 213-243 (1987). [ZBL0634.03047](https://zbmath.org/?q=an:0634.03047).
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As far as I am aware, this was the first proof of the consistency of $\mathfrak{r} < \mathfrak{s}$. Other models for this have been constructed since, for example in this recent paper by Guzman and Kalajdzievski.



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> *Guzmán, Osvaldo; Kalajdzievski, Damjan*, [**The ultrafilter and almost disjointness numbers**](https://doi.org/10.1016/j.aim.2021.107805), Adv. Math. 386, Article ID 107805, 41 p. (2021). [ZBL1493.03009](https://zbmath.org/?q=an:1493.03009).
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