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Minimal cardinality of non-bipartite sub-family of $[\omega]^\omega$

The cardinal $\mathfrak{nb}$ is equal to the reaping number $\mathfrak{r}$. An unsplit family is a collection $\mathcal R$ of infinite subsets of $\omega$ such that there is no set $D \subseteq \omega$...
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Given a polytope $P$ with bipartite edge-graph, if the bipartition classes are equal in size and lie on spheres, is $P$ inscribed?

After some simplifications, I arrived to the following general fact (it can be generalized even further). Paint the vertices in black and white according to the bipartite structure. We prove that, if ...
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