Kat\v{e}tov order between Hindman, Ramsey, van der Waerden and summable ideals

TL;DR

This paper compares the relative positions of Hindman, Ramsey, summable, and van der Waerden ideals within the Katetov order, revealing incomparability and non-boundedness among them in the context of set-theoretic ideals.

Contribution

It establishes the pairwise incomparability of Hindman, Ramsey, and summable ideals in the Katetov order and shows they are not below the van der Waerden ideal.

Findings

Hindman, Ramsey, and summable ideals are pairwise incomparable in the Katetov order.

These ideals are not below the van der Waerden ideal in the Katetov order.

The van der Waerden ideal contains sets with arbitrarily long arithmetic progressions.

Abstract

A family 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 I on X is below an ideal J on Y in the Katetov order if there is a function $f:Y\to X$ such that $f^{-1}[A]\in J$ for every $A\in I$. We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katetov 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. Moreover, we show that in the Katetov order the above mentioned ideals are not below the van der Waerden ideal that consists of those sets of natural numbers which do not contain arithmetic progressions of arbitrary finite length.