I SIGHED when they told me the cost of sponsoring the event.
The person sponsors a turkey roast on top of the mountain.
host a small event - an art opening, house party, etc.
sponsor a medium party or company event (like 催す（もよおす） but the emphasis is on 'sponsor' (i.e. provide money in exchange for publicity), where as もよおす has more to do with actually doing the work - arranging the chairs and snacks and sweeping the floor, and so on).
to repeat and repeat a demand. Can be used about corporations (This is your 3d notice of overdue library book) or between frends, ('I keep asking you to repay that $5 I lent you')
host a big event like the Olympics