This question assumes familiarity with combinatorial cardinal characteristics of the continuum.

Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing enumeration. Thus, for each natural number $n$, $a(n)$ is the $n$-th element of $a$ in increasing order. This way, every infinite set of a natural number is a member of $\mathbb{N}^{\mathbb{N}}$.

For $f,g\in\mathbb{N}^{\mathbb{N}}$, write $f\le^{\infty}g$ if $f(n)\le g(n)$ for infinitely many $n$. That is, if $g\not<^* f$.

For an infinite co-infinite set $a\subseteq\mathbb{N}$, let $a^{c}$ be its (infinite) complement in $\mathbb{N}$. Thus, $a$ and $a^{c}$ are both in $\mathbb{N}^{\mathbb{N}}$, and the following notion is natural.

A set $Y\subseteq\mathbb{N}^{\mathbb{N}}$ is *bi-nondominating*
if there is an infinite co-infinite set $a\subseteq\mathbb{N}$ such
that $Y\le^{\infty}a,a^{c}$. That is, for each $f\in Y$ we have
that $f\le^{\infty}a$ and $f\le^{\infty}a^{c}$.

**Definition.** $\mathfrak{bidi}$ is the minimal cardinality
of a set $Y\subseteq\mathbb{N}^{\mathbb{N}}$ that is *not* bi-nondominating.
That is, such that for each infinite co-infinite set $a\subseteq\mathbb{N}$,
there is $f\in Y$ such that $a\le^{*}f$ or $a^{c}\le^{*}f$.

It follows immediately that $\mathfrak{b}\le\mathfrak{bidi}\le\mathfrak{d}$. It can be shown that if we close the sequences in $Y$ under left shifts, then a witness for $Y$ being bi-nondominating actually reaps $Y$. Thus, $\mathfrak{bidi}\le\mathfrak{r}$, the reaping (or unsplitting) number. Since bounded sets in the Baire space $\mathbb{N}^{\mathbb{N}}$ are meager, and the operator $a\mapsto a^{c}$ is a hoemomorphism from the Cantor space $P(\mathbb{N})$ onto itself, we also have that the covering number for meager (or Baire first category) sets, $\mbox{cov}(\mathcal{M})$, is not greater than $\mathfrak{bidi}$. In summary, we have: $$ \max\left\{ \mbox{cov}(\mathcal{M}),\mathfrak{b}\right\} \le\mathfrak{bidi}\le\min\left\{ \mathfrak{r},\mathfrak{d}\right\} . $$

**Questions.** 1. Is $\mathfrak{bidi}$ a new combinatorial cardinal
characteristic of the continuum?

More precisely, is it conistently not in the lattice generated by the known combinatorial cardinal characteristics of the continuum by maxima and minima?

More concretely, are the inequalities $\max\left\{ \mbox{cov}(\mathcal{M}),\mathfrak{b}\right\} <\mathfrak{bidi}$ and $\mathfrak{bidi}<\min\left\{ \mathfrak{r},\mathfrak{d}\right\} $ each consistent?

**Guess.** My guess is that the answer (to all questions) is
"Yes", but I would be happier if it turns out to be "No."

**Motivation.** A topological space is *Menger* if for each
countable family of open covers of that space, there are finite subsets
of the open covers that together cover the space. This property was
introduced by Menger in 1924, and in this form by Hurewicz (1925).
It found numerous connections and applications to various branches
of mathematics, even to additive Ramsey theory.

A famous and notorious open problem asks whether, consistently, the
square of every Menger set of real numbers is Menger.
In a joint work with Piotr Szewczak, we proved that if $\mathfrak{bidi}=\mathfrak{d}$,
then there *is* a Menger set of real numbers whose square is
not Menger. Such a result was earlier known under the stronger hypothesis $\mbox{cov}(\mathcal{M})=(\mathfrak{d}=)\mbox{cof}(\mathcal{M})$.

**A survey of the standard models.** Following are comments about $\mathfrak{bidi}$ in models obtained by adding standard generic reals (usually, $\aleph_2$ many) to a model of CH.

*The Cohen model.* $\mbox{cov}(\mathcal{M})=\mathfrak{c}$, so $\mathfrak{bidi}=\mathfrak{c}$, too.

*The Random reals model.* $\mathfrak{d}=\aleph_1$, and thus $\mathfrak{bidi}=\aleph_1$.

*The Sack model.* Rarely a good model to separate cardinals of this kind, since all classic cardinals are $\aleph_1$ there.

*The Dominating reals (Hechler) model, the Laver model, the Mathias model.* $\mathfrak{b}=\mathfrak{c}$, so $\mathfrak{bidi}=\mathfrak{c}$.

*Miller's model.* $\mathfrak{r}=\aleph_1$, and thus $\mathfrak{bidi}=\aleph_1$.

Mildenberger suggests that adding Miller reals and then random reals may result in $\max\left\{ \mbox{cov}(\mathcal{M}),\mathfrak{b}\right\} < \min\left\{ \mathfrak{r},\mathfrak{d}\right\}$. What is $\mathfrak{bidi}$ in this model? (I guess that adding enough random reals to any model of $\mbox{cov}(\mathcal{M})<\mathfrak{d}$ wound render $\max\left\{ \mbox{cov}(\mathcal{M}),\mathfrak{b}\right\} < \min\left\{ \mathfrak{r},\mathfrak{d}\right\}$.)