Selective separability properties of Fréchet–Urysohn spaces and their products

We study the behaviour of selective separability properties in the class of Fréchet–Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Fréchet–Urysohn (hence $R$-separable and selectively separable) space which is not $H$-separable; assuming $\mathfrak p=\mathfrak c$, we construct such an example which is also zero-dimensional and $\alpha _{4}$. Also, motivated by a result of Barman and Dow stating that the product of two countable Fréchet–Urysohn spaces is $M$-separable under PFA, we show that the MA is not sufficient here. In the last section we prove that in the Laver model, the product of any two $H$-separable spaces is $mH$-separable.