For any $k$, the class of graphs of treewidth at most $k$ can be characterized by a finite set of forbidden minors.


For treewidth $1$ and $2$, the set is of size $1$. Then for treewidth $3$, the set is of size $4$, and beyond that the size of the set grows very rapidly. 


For a given $k$, is there a known upper bound on the number of vertices for the forbidden minors for treewidth at most $k$?



Size of forbidden minors for treewidth
Yes, an upperbound was proved in [Upper Bounds on the Size of Obstructions and Intertwines](http://ac.els-cdn.com/S0095895697917883/1-s2.0-S0095895697917883-main.pdf?_tid=78779910-7428-11e6-a86b-00000aab0f01&acdnat=1473162965_e2b1c9896d06ad788ff7004e3435616b) by Lagergren. The paper is behind a paywall, but the relevant theorem is Theorem 5.9.



> 
> If $G$ is an obstruction for graphs of tree-width at most $k$,
>  then $|E(G)|$ is at most doubly exponential in $O(k^5)
> $.
> 
> 
> 


In the same paper, Lagergren also bounds the size of a forbidden minor for path-width at most $p$. 



> 
> If $G$ is an obstruction for graphs of path-width at most
>  $p$, then $|E(G)|$ is at most exponential in $O( p^4
> )$.
> 
> 
> 


