Yi Zhang

We discuss a problem asked by E. van Douwen and A. Miller [5] in various forcing models

We compare several closely related continuum invariants, i.e., $\mathfrak{a}$, $\mathfrak{a}_\mathfrak{e}$, $\mathfrak{a}_\mathfrak{p}$ in two forcing models. And we shall ask some open questions in this field.

We discuss the cardinalities of maximal cofinitary groups under the assumption of $\neg CH$ . We also discuss various open questions in this area

We show that it is consistent with ZFC + ¬CH that there is a maximal cofinitary group (or, maximal almost disjoint group) G ≤ Sym(ω) such that G is a proper subset of an almost disjoint family A $\subseteq$ Sym(ω) and |G| < |A|. We also ask several questions in this area

We construct several forcing models in each of which there exists a maximal cofinitary group, i.e., a maximal almost disjoint group, G≤Sym, such that G is also a maximal almost disjoint family in Sym. We also ask several open questions in this area in the fourth section of this paper.

We show that it is consistent with ZFC + ¬CH that there is a maxima a most disjoint permutation family A ⊆ Symsuch that A is a proper subset of an eventually different family E ⊆ ℕℕ and |A| < |E|. We also ask severa questions in this area

In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofinitary group of Sym(ω) is strictly greater than the cardinal numbers o and a