Are you running Mint 12? Because Cinnamon won't be compatible with anything older, as it's based on Gnome 3.
Cinnamon is in the standard Mint 12 repository, no ppa needed.
Package name "cinnamon".
Edit: You're running it in Live mode, it isn't installed? Huh. Cinnamon wasn't in the Mint 12 repos when Lisa was released, it hadn't been developed yet. Have you refreshed the repositories (sudo apt-get update)? Cinnamon (and other new packages as well) might not show up in a Live session without manual refreshing things.
BTW, "MATE" isn't the default Mint 12 environment, Gnome Shell or MGSE is; double check to see if all the right desktops are available on your iso. Cinnamon depends on Gnome Shell, so if it isn't present that might be your culprit.
You can try installing it from the ppa:
sudo add-apt-repository ppa:gwendal-lebihan-dev/cinnamon-stable
sudo apt-get update
sudo apt-get install cinnamon
I'm posting this from Cinnamon on Ubuntu 11.10 using the above.