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Abstract. We say that an ideal I on ω is homogeneous, if its restriction to any I-positive subset of ω is isomorphic to I. The paper investigates basic properties of this notion -we give examples of homogeneous ideals and present some applications to topology and ideal convergence. Moreover, we answer questions related to our research posed in [1].

In this paper we study a new ideal $\mathcal{WR}$. The main result is the following: an ideal is not weakly Ramsey if and only if it is above $\mathcal{WR}$ in the Kat\v{e}tov order. Weak Ramseyness was introduced by Laflamme in order to characterize winning strategies in a certain game. We apply result of Natkaniec and Szuca to conclude that $\mathcal{WR}$ is critical for ideal convergence of sequences of quasi-continuous functions. We study further combinatorial properties of $\mathcal{WR}$ and weak Ramseyness. Answering a question of Filip\'ow et al. we show that $\mathcal{WR}$ is not $2$-Ramsey, but every ideal on $\omega$ isomorphic to $\mathcal{WR}$ is Mon (every sequence of reals contains a monotone subsequence indexed by a $\mathcal{I}$-positive set)

The main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets. This enables to transfer results concerning one direction to the other one.Consider a function g : ω → [0, ∞) such that limn→∞ g(n) = ∞ and n g(n)does not converge to 0. Then the family Zg = {A ⊆ ω : limn→∞ card(A∩n)= 0} is an ideal called simple density ideal (or ideal associated to upper density of weight g). We compare this class of ideals with Erdős-Ulam ideals. In particular, we show that there are ⊑-antichains of size c among Erdős-Ulam ideals which are and are not simple density ideals (in [12] it is shown that there is also such an antichain among simple density ideals which are not Erdős-Ulam ideals).We characterize simple density ideals which are Erdős-Ulam as those containing the classical ideal of sets of asymptotic density zero. We also characterize Erdős-Ulam ideals which are simple density ideals. In the latter case we need to introduce two new notions. One of them, called increasing-invariance of an ideal I, asserts that given B ∈ I and C ⊆ ω with card(C ∩ n) ≤ card(B ∩ n) for all n, we have C ∈ I. This notion is inspired by [3] and is later applied in [12] for a partial solution of [15, Problem 5.8].Finally, we pose some open problems.

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For any Borel ideal we characterize ideal equal Baire system generated by the families of continuous and quasi-continuous functions, i.e., the families of ideal equal limits of sequences of continuous and quasi-continuous functions

This paper studies the combinatorics of ideals which recently appeared in ergodicity results for analytic equivalence relations. The ideals have the following topological representation. There is a separable metrizable space X, a σ-ideal I on X and a dense countable subset D of X such that the ideal consists of those subsets of D whose closure belongs to I. It turns out that this definition is indepedent of the choice of D. We show that an ideal is of this form if and only if it is dense and countably separated. The latter is a variation of a notion introduced by Todorčević for gaps. As a corollary, we get that this class is invariant under the Rudin-Blass equivalence. This also implies that the space X can be always chosen to be compact so that I is a σ-ideal of compact sets. We compute the possible descriptive complexities of such ideals and conclude that all analytic equivalence relations induced by such ideals are Π 0 3 . We also prove that a coanalytic ideal is an intersection of ideals of this form if and only if it is weakly selective.

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