This question concerns a new cardinal characteristic of the continuum that arose out of issues in my answer to the question, Sunflowers in maximal almost disjoint families.

A family $\cal A$ of infinite subsets of $\omega$ is *almost
disjoint*, if any two members of the family have finite
intersection. Such a family is a *maximal almost disjoint family*
if it cannot be extended to a larger almost disjoint family.

A *$\Delta$-system*, also known as a *sunflower*, is a family of
sets with all pairs having the same pairwise intersection.

In his earlier question, Dominic had asked whether every maximal almost disjoint family must contain an infinite sunflower. In the general case, this seems still to be open, but my answer there shows that under the continuum hypothesis, there is a maximal almost disjoint family containing no sunflowers even of size $3$. Indeed, the construction there shows that there is a maximal almost disjoint family $\langle A_\alpha\mid\alpha<\omega_1\rangle$ such that every $A_\alpha$ has different intersections with every earlier $A_\beta$, for $\beta<\alpha$. This property implies that there can be no sunflower of size $3$ in this family. (But notice by a simple pigeon-hole argument that it will be impossible to extend this stronger property to enumerations beyond $\omega_1$.)

My questions concern the property of almost disjoint families that are maximal with respect to the property of not containing any sunflower of a certain size.

**Question 1.** If an almost disjoint family of infinite subsets of
$\omega$ is maximal amongst almost disjoint families with respect
to the property of not containing a sunflower of size $3$, is it a
maximal almost disjoint family?

And more generally, I ask the same for sunflowers of any particular size.

The question leads naturally to new cardinal characteristics of the
continuum. Namely, let us define *almost-disjointness sunflower
number*, officially denoted $\frak{a}_{\kappa}^\Delta$, but let me immediately drop the superscript and write just $\frak{a}_\kappa$, to be the size of the smallest
almost-disjoint family that is maximal among almost-disjoint
families with respect to the property of not containing a sunflower
of size $\kappa$. (We consider only $\kappa\geq 3$.)

The construction on my other answer shows that $\omega_1\leq\frak{a}_{\kappa}$.

**Question 2.** Can we separate these various cardinal characteristics $\frak{a}_\kappa$ from each other, and from the almost-disjointness number $\frak{a}$?

For example, is it consistent with ZFC that $\frak{a}_{3}<\frak{a}$? This would be a strong refutation of question 1. Is it consistent that $\frak{a}_{3}\neq\frak{a}_{4}$?

At first I had though it was clear that $\frak{a}_{\kappa}\leq\frak{a}$, the almost-disjointness number, which is the smallest size of any maximal almost disjoint family. But upon reflection, this no longer seems clear to me, since perhaps there could be a small maximal almost disjoint family, but it contains a lot of sunflowers, and the smallest maximal sunflower-free family might be larger. Or strictly smaller, since a maximal sunflower-free family might not be a maximal almost disjoint family. I had similarly expected that if $\kappa<\lambda$, then there should be some trivial provable relation between $\frak{a}_{\kappa}$ and $\frak{a}_{\lambda}$. But unless I am mistaken, this now also doesn't seem to be immediate.

**Question 3.** What are the provable relations between $\frak{a}_\kappa$, $\frak{a}_\lambda$, and $\frak{a}$ when $\kappa<\lambda$?

Can we say something even about the relation of $\frak{a}_{3}$ and $\frak{a}_{4}$, or their relation to $\frak{a}$?