math.LO — Logic

The random graph ideal and fractional coloring

Contributed by Carlos López-Callejas

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 graph ideal is proper and is described by the fractional cochromatic numbers of its finite induced subgraphs.