Answer:
No. If a collegue hire 7 math Ph.D.s, at least 2 will be from the same university.
The only way to have a job for all of them is that all colleges hire 6 math Ph.D.s, one from each university.
Step-by-step explanation:
If the condition is that <em>no college will hire more than one Ph.D. from any given university, </em>no college can hire more than 6 math Ph.D.s, as there are only 6 universities to choose from. If they hire seven, they have to repeat some university, violating the condition.
Then, the only way to have a job for all of them is that all colleges hire 6 math Ph.D.s, one from each university.