TITLE: Finitely generated group every 2-generated subgroup of which is finite
QUESTION [9 upvotes]: I know of Tarski monsters and the Burnside Problem. I would like to know if there is an infinite finitely generated group $G$ such that for any $g$ and $h$ in $G$, the subgroup generated by $\{g,h\}$ is finite.
I am also interested in related questions:

For which $(m,n)$ does there exist an infinite $m$-generated group every $n$-generated subgroup of which is finite?
For which $(m,n,k)$ does there exist an infinite $m$-generated group every $n$-generated subgroup of which has cardinality at most $k$?

Thank you.

REPLY [9 votes]: Questions 1 and 2 are answered by Golod's theorem, see Theorem 3.3 in 
the survey:
M. Ershov, Golod-Shafarevich groups: a survey. Internat. J. Algebra and Computation, vol. 22 (2012). 
Namely, for every $d\ge 2$ there exists an infinite $d$-generated group such that every $d-1$-generated subgroup is finite. 
(He even gets a finite $p$-group for the given prime number $p$ but this is irrelevant.) 
As for the last question, I am not sure, most likely it is unknown. I suggest to ask Ershov directly.