Would a Pankration fighter be allowed in the UFC? If so, how would he be able to fend off other foes who use Karate, Muay Thai and Krav maga?
http://ancientolympics.arts.kuleuven.be/eng/TC007cEN.html claims of Pankration:
everything was permitted except biting, gouging (stabbing with your finger in your opponent's eye, nose or mouth) and attacking the genitals.
If that's accurate, then you couldn't compete in the UFC and use the full set of pankration techniques. There are other questions listing the rules for UFC, or google turns it up easily.
That said, anyone from any style who abides by the UFC rules should be able to compete in UFC with the subset of their techniques that are legal. (You're not going to get straight into a major event without some kind of fame or qualifying success though.) And, if some technique that wasn't explicitly banned before proved crazily injurious, they might ban it too. There's plenty of scope to argue about which techniques should be banned... just as one example - I think if people run at you with their head down wanting to take you down, they're inviting a downward elbow to the back of the head, neck or spine, and a set of rules that artificially protects them from those strikes while they have a crack at a takedown undermines the whole claim of UFC to be a more realistic way to see how striking and grappling interact.
Pankration was a sporting event introduced into the Greek Olympic Games in 648 BC and was an empty-hand submission sport with scarcely any rules. The athletes used boxing and wrestling techniques, but also others, such as kicking and holds, joint-locks and chokes on the ground making it similar to modern MMA.
Just that they'd be subject to the same rules as any other UFC fighter.
Personally, when I watch McGreggor & St-Pierre, I see pankration roots in their striking in particular.
(Comparison is vase paintings, which can have surprising level of detail, but it's a good bet that boxing, grappling, and other techniques have been practiced in some form continuously in Europe since at least the Greek and Roman era.)