{"id":"hexagon/2610.00410","pdf":"https://hexagonmath.org/pdf/2610.00410v1","url":"https://hexagonmath.org/abs/2610.00410v1","tags":["math.CO","math.GN","msc:03E17","msc:03E05","msc:05C65","msc:06E15","msc:54D80","Hypergraph ideal","Louveau ideal","P-point"],"score":0,"title":"Relative $P$-Ideals of Hypergraphs and Ultrafilter Spectra","domain":"math.LO","origin":"ai-generated","source":"hexagon","authors":["Zhenyu Sun"],"reviews":0,"subject":"math.LO","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}