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Let $V$ be a model of set theory with CH. Recall the following definitions.

A forcing $\mathbb{P}\in V$ has the Laver property if for any $\mathbb{P}$-generic filter $G$ over $V$, functions $f\in\omega^\omega\cap V$ and $g\in\omega^\omega\cap V[G]$ such that $g\le_*f$, there exists a function $G\in\big([\omega]^{<\omega}\big)^\omega\cap V$ such that $g(n)\in G(n)$ and $|G(n)|\le n+1$ for every $n\in\omega$.

A forcing $\mathbb{P}\in V$ preserves reals non-meager if $\mathbb{R}\cap V$ is a non-meager subset of $\mathbb{R}\cap V[G]$ for any $\mathbb{P}$-generic filter $G$ over $V$.

Typical examples of proper forcings having the Laver property and preserving reals non-meager are Sacks and side-by-side products of Sacks, Miller and Silver (-like). The properties are also preserved by countable support iterations.

Question 1: What are other (natural) examples of proper forcings with the Laver property and preserving reals non-meager?

Obviously, if $\mathbb{P}$ preserves reals non-meager, then the left half of the Cichoń's diagram must be $\omega_1$ in the extension. However, e.g., in the countable support iteration of length $\omega_2$ of the Miller forcing one gets that $\mathfrak{d}=\omega_2=\mathfrak{c}$ and of a Silver-like forcing that $\mathfrak{r}=\omega_2=\mathfrak{c}$.

Question 2: What other standard cardinal characteristics of the continuum can we make big using a proper forcing having the Laver property and preserving reals non-meager?

E.g. what about $\mathfrak{a}$, $\mbox{cof}(\mathcal{M})$ or $\mbox{cof}(\mathcal{N})$ (while $\mathfrak{d}=\omega_1$) etc.?

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Somewahat relevant might be Chapter 5 in "Norms on possibilities I: forcing with trees and creatures" by A. Roslanowski and S. Shelah, Memoirs of the American Mathematical Society, 141, 1999.

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  • $\begingroup$ Thanks, I'll look into it. (Btw, someone provided this reference 2 hours ago but deleted his/her comment...) $\endgroup$ – Damian Sobota Oct 9 '17 at 0:32
  • $\begingroup$ Dear Andrzej, do you maybe know any answer for the second question? $\endgroup$ – Damian Sobota Oct 9 '17 at 18:35
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I might be incorrect, but I think there is a subfamily of usable creature forcings which have the Laver property.

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    $\begingroup$ Any examples or references? $\endgroup$ – Damian Sobota Oct 8 '17 at 21:06

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