The meaning of some asymptotic notation

Question

I am trying to make sense of some notation in a paper I am reading. In particular, in the following excerpt, I've circled the notation that I do not understand:

This is a restatement of a theorem in an earlier paper in which this particular notation does not appear, but rather it is assumed that $k \ll t \ll v$. So my guess is that $t \geq t_o(k)$ captures $k \ll t$ and $v \geq v_o(k,t)$ captures $t \ll v$ and $k \ll v$, though I'm not too sure. What is the correct way of interpreting this kind of notation?

Answer 1

It's not very clear, but I would interpret it as meaning that there exists some $t_0$ depending only on $k$ such that for every $t\geq t_0$ we can find $v_0$ (depending on both $t$ and $k$) such that the statement is true for every $k$-uniform hypergraph with maximum degree bounded by $t^{k-1}$ and at least $v_0$ vertices.

This is also how I would interpret $k\ll t\ll v$ assuming that it's clear that the dependency goes in that direction¹ and the person who wrote it is doing combinatorics - I think that could mean something quite different in other areas of mathematics.

¹ that is, as opposed to $v$ being the basic quantity with $t$ and $k$ having to be sufficiently small.