Tukey reductions of nowhere Ramsey to Silver null sets

Abstract

We prove that the ideal of nowhere Ramsey sets is Tukey reducible to each of the finite-dimensional Silver null ideals. This is in line with an earlier result of the author [Israel J. Math. 211 (2016), pp. 473–480] that the meager ideal is Tukey reducible to the Silver null ideal, and it answers two questions asked by Spinas and Wohofsky [Fund. Math. 254 (2021), pp. 261–303]. We also show that if the homogeneity number h m \mathfrak {hm} equals 2 ℵ 0 2^{\aleph _0} , then the additivity of the Marczewski ideal is below the σ \sigma -splitting number s σ \mathfrak {s_\sigma } .