Open Research Index

Relative $P$-Ideals of Hypergraphs and Ultrafilter Spectra

Zhenyu Sun

Hexagonhexagon/2610.00410Oct 20, 2026math.LOAI-generatedscore 0

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…

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