Question: I emailed a potential supervisor at a US university recently. He replied promptly, cc'ing the graduate school dean in the email and saying I had a very good idea but then suggesting another research question that needs to be answered first. 

I saw that no one has done it yet (nothing published, as far as I know), and I then asked if his lab is currently tackling the problem, and he simply responded:

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> We are not working on it yet
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I have asked the graduate school dean several questions after that, but I have not responded to the potential supervisor yet. How should I continue our correspondence? I'm really interested in the new research problem, but I don't know how to respond to him.

Answer: 
I think he is busy and just responded in the quickest way possible to your query. If you are interested in it, respond that you are and would like to take it up. More importantly, respond that you want to take it up under his supervision and will be following up with your formal application. 

If you come up with any deep thoughts on the problem or partial solutions, you can (now or later) communicate that you have progress and maybe share some of that. Since you don't really know much about him, I wouldn't lay everything out early on, however. Wait until you can assess one another first. 

If you read other questions on this site note that some of them are about unethical behavior by supervisors. I have no reason to even suspect this is the case here (and since it is rare, I doubt it). Others questions are about misunderstandings. If you are familiar with the game [Poker](https://en.wikipedia.org/wiki/Poker), don't show all your cards early.
One thing is, be clear and specific. If you are looking for a place to do graduate study, be sure to ask for this. "I am looking for a place to do my PhD. Would it be possible for me to be a PhD student at your lab? Would you be available to be my supervisor to complete my PhD?" If the answer is hazy, or unclear, or non-committal, then consider that you will soon need to make clear commitments to apply to a university. You may need to pay out money to apply. You may have to forego other possibilities to apply, due to time and money.

His response might reasonably be to ask for your record so far, like transcripts or references etc. And if you have financial support of some kind. And all the mundane things.

If there is agreement to take you as a PhD candidate, but the specific research topic is not decided in advance, you need to decide if that is a deal-killer for you. Again, ask specific questions. "What possible research topics would you be willing to supervise me for?" Again, if the answers don't please you, or they are vague, you need to regroup and re-plan.

With luck your academic career isn't limited to one research question. So, if you don't do the first choice question now, you may be able to keep that "in your pocket" for future work. That's got some value. It is a rare PhD candidate who gets to do all the research stuff he wants without fitting in to some existing research program. And being able to "hit the ground running" with your own original ideas after you get the PhD is pretty good.

Any reasonable prof will understand that you need to make firm decisions so you can decide what school you are attending. And who is to be your supervisor. If he won't, or can't, be helpful on that, maybe he is not the right supervisor. Possibly it is not that he is a bad person. Maybe he is just busy and distracted. But "busy and distracted" now, while it is your only data, is still data.