math.LO — Logic

Relative $P$-Ideals of Hypergraphs and Ultrafilter Spectra

Contributed by Zhenyu Sun

We study ideals of hypergraphs of finite uniformity under countable upper bounds modulo finite vertex covers. The canonical ideal generated by a free filter and finite weak colorings has this relative $P$-property exactly when the filter is the intersection of finitely many $P$-points. The classification is preserved by adjoining hypergraphs that are locally finite after a finite vertex deletion. The corresponding extension of a trace ideal induces capture ideals on increasing sequences. These are $\sigma$-ideals exactly for $P$-filters, and their uniformities are $\max\{\pi\chi(\mathcal{F}),\mathfrak{d}\}$. For an ultrafilter $\mathcal{U}$, the arity-two capture ideal equals the Louveau ideal exactly when $\mathcal{U}$ is a $P$-point. Successive capture ideals decrease with the arity, and equality characterizes $Q$-points. Every strict inclusion is witnessed by a closed set. Thus a non-$Q$ $P$-point yields a decreasing sequence of $\sigma$-ideals, distinguished by closed sets, with equal uniformities. We also establish invariance under locally finite witness enlargements and analyze Stone kernels and countable block products. Consistently, nested ray-positive $P$-sets have the same singleton endpoint spectrum and different ray transversal numbers.