How do I give my graduate a deadline for the job hunt?

This can be a real time of bonding or contention. Before the graduate comes home talk candidly about what the expectations will be. Set a 4 month deadline with conditions. Agree together on what household responsibilities your grad will have to earn his “keep.” Ideas are cleaning, cooking, laundry, mowing lawn. Be clear 4 months of free room and board are all you are willing to allow before starting to charge rent. I suggest you agree that the graduate get a part-time job immediately such as driving Lyft or Favor, or working at Rudy’s or Dairy Queen (both pay $14 an hour!), etc. Because honestly there needs to be downtime from applying and writing resumes. I suggest they commit to doing 2 - 3 hours three times a week of job application work and then go do other things, including work part-time. Put your agreement in WRITING. This will help avoid misunderstandings AND fights. You can always renegotiate the agreement if you see fit.