Will Synergy recognize or allow four clients, one of which has 3 monitors for a total of 6 monitors?
It will work, but how you configure it depends on the OS. Under Windows, all attached monitors are treated as one "screen", so depending on the layout multiple screens may share an edge (so if you have two screens side by side, there is only one "top" edge. If you want to split this so that moving up on either screen goes to a different monitor, you will need to edit the config file and use partial edges.)
Under Linux with Xinerama (or nVidia TwinView) this is the same, but if you use separate X11 screens (aka Zaphod mode, where you can't drag windows between monitors, and DISPLAY=:0.1, :0.2, etc.) then it behaves quite differently and you may need to run one instance of Synergy for each screen.
I'm not sure about a multi-monitor Mac, but I'm guessing it's the same as Windows and Linux with Xinerama.
This is assuming the monitors form a more-or-less complete rectangle with no gaps (e.g. all the screens are in a line.) If the PC with three screens doesn't have them in a line (e.g. two side by side and one above) then it becomes much more complex and may not be possible to switch screens from every monitor edge you would like to.
I use it with one Windows machine with 2 monitors, one Ubuntu machine with two monitors, and one Ubuntu machine with 1 monitor. I have added a 4th before, but didn't use it so got rid of it. So I believe you would be fine with that setup.
My Windows PC is the server, as setting up the server in linux is painful at best. (at least to me)
AFAIK there is no limitations on Synergy's support; as long as you are able to adequately describe the layout in the configuration, its possible. I ran the server on a dual-display iMac with a client being a triple-display Windows machine before, so I would guess that you'd be fine.
Yes, I've done it before with non-asymmetric number of monitors, and it worked great!
I have gotten it to work on 3 monitors, two on an Ubuntu 12.04 host, and one on a Windows 10 host.
I'm using the Ubuntu host as the server because I am docking my Windows 10 Notebook for use with a monitor.
I first started with one monitor each, configured the Synergy server on my Ubuntu host like this:
Next, I plugged in a second monitor on my Ubuntu host. When I plugged it in, I noticed my display changed to this:
Oddly, I was only able to toggle between my Ubuntu displays - I could not access my Windows Display. The mouse would leave the right-hand side of Ubuntu Display 1, and enter on the far left of Ubuntu Display 2. I only had one instance of Synergy running and it was still configured with Ubuntu on the left and Windows on the right.
So, I simply swapped the cables on my monitors and then got:
While the mouse now left Ubuntu Display 1 and smoothly transitioned to Ubuntu Display 2, my Windows 10 desktop was still inaccessible.
Having read some of the comments above, out of sheer curiosity, I launched a second instance of Synergy on my Ubuntu Box (remember, I chose this as my server) and 'voila!' - now I am able to transition through all three displays the way I was hoping.
Keeping this in mind, I am thinking that you'll have to do the same on your setup, and run one instance for every additional screen you want to add.
While I really can't say if that will work, I was hoping that documenting this would help someone out.