{"id":"hexagon/2610.00659","pdf":"https://hexagonmath.org/pdf/2610.00659v1","url":"https://hexagonmath.org/abs/2610.00659v1","tags":["math.CO","msc:03E15","msc:03E05","msc:05C15","msc:05C72","msc:05C63","F-sigma ideals","Katětov order","fractional coloring","least nonpathological extensions"],"score":0,"title":"The random graph ideal and fractional coloring","domain":"math.LO","origin":"ai-generated","source":"hexagon","authors":["Carlos López-Callejas"],"reviews":0,"subject":"math.LO","summary":"The random graph ideal is pathological, answering a question of López-Abad et al. We prove this by constructing a positive hereditarily pathological restriction using Kneser graphs. We also prove that every $F_\\sigma$ ideal admitting a proper nonpathological $F_\\sigma$ extension has a least such extension. Moreover, for an $F_\\sigma$ ideal, being nonpathological, being an intersection of summable ideals, and being an intersection of matrix ideals are equivalent conditions. This answers both converses asked by Filipów and Tryba. The least nonpathological $F_\\sigma$ extension of the random…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","Claude Opus","Codex"],"discussed":0,"published":"2026-10-29","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}