Definition: The subgroup rank of a finite group $G$ is the minimal natural number $n$ such that every subgroup of $G$ can be generated by $n$ elements (or fewer).
This invariant has been studied extensively for various families of groups. I am interested in the family of finite simple groups and I have been unable to find and relevant information in the literature.
Question 1: Are there only finitely-many finite simple, non-abelian groups $G$ of a given subgroup rank $n$?
Some relatively straight-forward comments and reductions:
It is not too difficult to show that there are only finitely-many alternating groups of subgroup rank at most $n$ (by explicitly constructing elementary-abelian subgroups of a certain subgroup rank). There are also only finitely-many sporadic groups, according to the classification. These observations reduce the above question to finite simple groups of Lie-type.
Question 2: Are there only finitely-many finite simple groups $G$ of Lie-type with given subgroup rank $n$?
It is again not too difficult to show that the "field rank" of $G$ is bounded from above by a function of $n$ (by looking at the natural homomorphism from the field to the root subgroups). It is also possible to show that the Lie-rank of $G$ is bounded from above by a function of $n$. These observations further reduce Question 2 to bounding the defining characteristic of the simple group of Lie-type by some function that depends only on the subgroup rank $n$. Unfortunately, I do not have any good intuition to determine whether the latter statement is true or not.
I hope both questions have a positive answer because that would give us a nice property about the FSG. But I suspect we can prove the answers to be "no" by simply making some judicious choice for the Lie-type, field-rank, and Lie-rank and by then looking at the structure of the Sylow-subgroups of $G$, as the characteristic goes through the different primes.