We present two ${\mathbb{P}}_{max}$ varations which create maximal models relative to certain counterexamples to Martin’s Axiom, in hope of separating certain classical statements which fall between MA and Suslin’s Hypothesis. One of these models is taken from [19], in which we maximize relative to the existence of a certain type of Suslin tree, and then force with that tree. In the resulting model, all Aronszajn trees are special and Knaster’s forcing axiom ₃ fails. Of particular interest is the still open question...

By an equivalence system is meant a couple $𝒜=(A,\theta )$ where $A$ is a non-void set and $\theta$ is an equivalence on $A$. A mapping $h$ of an equivalence system $𝒜$ into $ℬ$ is called a class preserving mapping if $h\left({\left[a\right]}_{\theta }\right)={\left[h\left(a\right)\right]}_{\theta {}^{\text{'}}}$ for each $a\in A$. We will characterize class preserving mappings by means of permutability of $\theta$ with the equivalence ${\Phi }_h$ induced by $h$.

We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a $T_{31/2}$-space. In [9] we showed that $C_p\left(X\right)$ has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. $C_p\left(X\right)$ has countable fan tightness and the Reznichenko property. 2....