The best advice is to check with your sponsor.

The second best advice is to do some research on your own. Go to flight status on United.com and look at the seat maps for every flight your are interested in.

For example, yesterday's UA970, ORD-FCO, went out with no empty seats. UA42, IAD-FCO, same deal. UA944 ORD-FRA had one seat. but you'd have to make it to Rome by train. And so on. After a few weeks, you'll have a better idea of airport, day of week, and flight.