Is expiation obligatory for someone who said to his friend, "By Allah, if you do such and such, I will do such and such," and then did not do what he swore to do after his friend did that?

You are not required to offer expiation for an oath (kaffarat yamin) as long as you have not specified a definite time for performing the action sworn upon, unless you despair of performing it. So, if you swore to do something and did not specify a time for it, you do not break your oath until you despair of performing it, such as if the object sworn upon is destroyed or the one who swore dies. This is because no specific time was set, and the action is possible at any time, so the violation is not realized except through despair.