The Stacks project

As worded, the definition sounds as if it is requiring the finite maximal chains to have equal length, while imposing no condition on infinite maximal chains. I suspect that the definition should instead say

A ring $R$ is said to be catenary if for every pair of prime ideals $\mathfrak{p} \subset \mathfrak{q}$, there exists an integer bounding the lengths of all finite chains of primes $\mathfrak{p} = \mathfrak{p}_0 \subset \mathfrak{p}_1 \subset \cdots \subset \mathfrak{p}_e = \mathfrak{q}$, and the maximal such chains all have the same length.

Less importantly, I wonder also whether it would be clearer to define "saturated" chain and to use that word instead of "maximal" since I could imagine that some people could misunderstand maximal to mean maximal-length (even though they would quickly realize that that could be not be what was meant).

Even less important: Note that it is better to use \cdots instead of \ldots in the chain.

OK, I changed the definition as you suggested. To make sure I added a definition of "chain of primes" in a ring because this is used all over the place. (The definition was implicit in the definition of the dimension of a ring.) See changes here.

Side comment: I dislike cdots and I like ldots, so I consistently use ldots for everything.