According to the introduction in

<cite authors="Cooper, D.; Thurston, W. P.">_Cooper, D.; Thurston, W. P._, [**Triangulating 3-manifolds using 5 vertex link types**](http://dx.doi.org/10.1016/0040-9383(88)90004-3), Topology 27, No. 1, 23-25 (1988). [ZBL0656.57004](https://zbmath.org/?q=an:0656.57004).</cite>

>It is known that, for any dimension $n$, there is a finite set of link types such that every $n$-manifold has a triangulation in which the link of each vertex is in this set.

(I assume the statement is about PL manifolds.) 

What is a proof of or a reference to this *known* result?

Edit. There is a related discussion by Florian Frick [here](http://pi.math.cornell.edu/~frick/valence_bounds.pdf), but he only gets a bound on the valence of "ridges" (codimension 2 faces). I do not see how to generalize his sketch to faces of higher codimension.  (Actually, I cannot follow his sketch.)