We are preparing to implement SSO with ADFS for our office 365 implementation but still have some confusion about the number of certificates needed.

In microsoft's recommended configuration you should have 4 servers for high availability, 2 ADFS servers (internal network) and 2 ADFS proxy servers (in DMZ accessible from internet).

I understand you need a 3rd party SSL cert installed when you create your first ADFs server. Do you then need additional certs for the second ADFS server or proxy servers?