logical deduction - Five professors and nine dishes

Here's yet another puzzle adapted from a puzzle book (in my case, with edits to make some of the specifications of the puzzle more clear because I didn't really understand them the first time I read it).

Five professors are spending three days on a retreat, and every day they eat dinner at the same restaurant. This restaurant has a special "roulette menu" with nine items, in which you choose a menu item from 1 to 9 that corresponds to one of the nine available items, but you don't know which number corresponds to which item. When the items are ordered, they are brought to the table in no particular order, so if a group orders multiple items at a time, they cannot figure out individually which item was ordered by whom.

These professors have tasked themselves with figuring out which number corresponds to which item in three days regardless. Each professor will order one item per day. How should they plan their ordering of the items in order to find out which number corresponds to which item for all items?

Answer

Okay, 15 orders, 9 dishes, that means 6 dupes. So 3 dishes will only be ordered once, obviously never on the same day. Now to split the six dupes so they can all be detected.

The first 3:

Day 1: 1   3
Day 2: 1 2
Day 3:   2 3

The dish that shows on day 1 and 2 is #1, and so on.

Now add in same-day pairs:

Day 1: 1   3 4 4
Day 2: 1 2   5 5 
Day 3:   2 3 6 6

So the dish that's doubled on day 1 is #4, etc.

And now the last three:

Day 1: 1 7 3 4 4
Day 2: 1 2 8 5 5 
Day 3: 9 2 3 6 6

The dish that only shows up on day 1 is #7, and so on.

And then, secure in their triumph, they can leave five bad Yelp reviews warning people away from this crazy place.