Katětov order between Hindman, Ramsey and summable ideals

Abstract

A family \(\mathcal {I}\) of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal \(\mathcal {I}\) on X is below an ideal \(\mathcal {J}\) on Y in the Katětov order if there is a function \(f{: }Y\rightarrow X\) such that \(f^{-1}[A]\in \mathcal {J}\) for every \(A\in \mathcal {I}\). We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where

• The Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph),

• The Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory),

• The summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent.