# What do these lower bounds really mean?

I was reading an abstract for a paper that gives a minimum number M of good agents that guarantees f-Byzantine gathering, if there are f Byzantine agents. It gives a lower bound of f+1 for strongly byzantine agents. It also says that:

"...we give deterministic gathering algorithms for at least 2f+1 good agents when the size of the network is known...".

There are two lower bounds to the number of good agents. What do those two lower bounds mean? If it was only one lower bound, it would have been clear.

The abstract is: Gathering Despite Mischief

So for $f$ weakly Byzantine agents any number of good agents will do if the network size is known. Alternatively if it is not known, $\ge f+2$ good agents is a lower bound and they also show the matching upper bound.
In the strongly Byzantine case, they show that $f+1$ is a lower bound on the number of good agents required when the network size is known, which implies that this is true when the network size is unknown. These lower bounds aren't tight it seems as their best algorithms require $\ge 2f+1$ good agents in the case where the network size is known, or $\ge 4f+2$ good agents if the network size is unknown.