Fastest possible spread of illness at a dinner

At a banquet of $1000$ guests, one guest is ill and the rest are well. Every minute each guest chats with exactly one other guest, and repeat conversations are allowed. Whenever a well guest chats with an ill guest, the well guest becomes ill and stays ill. What is the shortest possible time, in minutes, before all $1000$ guests are ill?

10. Each ill guest can infect at most one new person per minute, so the number ill can at most double each minute: after $n$ minutes at most $2^n$ are ill. Since $2^9=512<1000\le1024=2^{10}$, ten minutes is the minimum, and perfect pairing achieves it.