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.



Finitely generated group every 2-generated subgroup of which is finite
Questions 1 and 2 are answered by Golod's theorem, see Theorem 3.3 in 
the survey:


M. Ershov, [Golod-Shafarevich groups: a survey](https://arxiv.org/pdf/1206.0490v1.pdf). 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.


