Chasing up response to PhD offer I'm asking this from the perspective of the future supervisor. After a round of interviews I approached the highest ranked candidate. My error was not including a deadline in the initial email. 

Now a week has passed from the last response I got from the candidate. I have sent another reminder recently. However, my patience is running out. I understand that of course the candidate might be trying to buy some time to make up their mind, and probably waiting for other offers.

Are there any other options apart from sending an ultimatum? If so, how much longer should I wait to filter out external reasons for the lack of responses? I.e. sickness, unforeseen events, etc.? This is in Europe.
Do you have a phone number for the candidate?

If you contact them by phone you will know whether or not they received your communication, rather than email where, as you say, you don't know if they may have experienced extenuating circumstances preventing them from responding.

If they are indeed waiting for other responses, they might be more likely to mention this on the phone rather than in an email.