Katětov order between Hindman, Ramsey and summable ideals

A family \(\mathcal {I}\) $\mathcal{I}$ 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 \(\mathcal {I}\) $\mathcal{I}$ on X is below an ideal \(\mathcal {J}\) $\mathcal{J}$ on Y in the Katětov order if there is a function \(f{: }Y\rightarrow X\) $f:Y\to X$ such that \(f^{-1}[A]\in \mathcal {J}\) $f^{-1}[A]\in \mathcal{J}$ for every \(A\in \mathcal {I}\) $A\in \mathcal{I}$ . 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),