Cardinal invariants of monotone and porous sets

Abstract

A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y)≤ c d(x,z) for all x<y<z in X. We investigate cardinal invariants of the σ-ideal Mon generated by monotone subsets of the plane. Since there is a strong connection between monotone sets in the plane and porous subsets of the line, plane and the Cantor set, cardinal invariants of these ideals are also investigated. In particular, we show that non(Mon)≥𝔪_σ-linked, but non(Mon)<𝔪_σ-centered is consistent. Also cov(Mon)<𝔠 and cof(𝒩)<cov(Mon) are consistent.