Given a family${\cal C}$of infinite subsets of${\Bbb N}$, we study when there is a Borel function$S:2^{\Bbb N} \to 2^{\Bbb N} $such that for every infinite$x \in 2^{\Bbb N} $,$S\left \in {\Cal C}$and$S\left \subseteq x$. We show that the family of homogeneous sets as given by the Nash-Williams’ theorem admits such a Borel selector. However, we also show that the analogous result for Galvin’s lemma is not true by proving that there is an$F_\sigma $tall ideal on${\Bbb N}$without a Borel selector. The (...) proof is not constructive since it is based on complexity considerations. We construct a${\bf{\Pi }}_2^1 $tall ideal on${\Bbb N}$without a tall closed subset. (shrink)

We study the Katětov order on Borel ideals. We prove two structural theorems, one for Borel ideals, the other for analytic P-ideals. We isolate nine important Borel ideals and study the Katětov order among them. We also present a list of fundamental open problems concerning the Katětov order on Borel ideals.

We address some phenomena about the interaction between lower semicontinuous submeasures on $${\mathbb {N}}$$ N and $$F_{\sigma }$$ F σ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $$F_{\sigma }$$ F σ ideals. We give a partial answers to the question of whether every nonpathological tall $$F_{\sigma }$$ F σ ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological $$F_{\sigma }$$ (...) F σ ideals using sequences in Banach spaces. (shrink)

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

We continue with the study of the Katětov order on MAD families. We prove that Katětov maximal MAD families exist under $\mathfrak {b=c}$ and that there are no Katětov-top MAD families assuming $\mathfrak {s\leq b}.$ This improves previously known results from the literature. We also answer a problem form Arciga, Hrušák, and Martínez regarding Katětov maximal MAD families.

We study the following natural strong variant of destroying Borel ideals: $\mathbb {P}$ $+$ -destroys $\mathcal {I}$ if $\mathbb {P}$ adds an $\mathcal {I}$ -positive set which has finite intersection with every $A\in \mathcal {I}\cap V$. Also, we discuss the associated variants $$ \begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y| \omega $ ; (4) we characterise when the Laver–Prikry, $\mathbb {L}(\mathcal {I}^*)$ -generic real $+$ -destroys $\mathcal {I}$, and in the case of P-ideals, when exactly $\mathbb {L}(\mathcal {I}^*)$ $+$ -destroys $\mathcal {I}$ ; (...) and (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal. (shrink)