Everybody present

Question

Once upon a time there was a big party with n participants. People arrived to the party,spent some time there and went home. the bartender noticed that any two people had a drink together.
a)Show that there was a moment when everybody was present at the same time
b)What should we assume about endpoints of the time intervals for the statement to hold?
c)What if there were infinitely many people?

My approach:

Since everyone had a drink together then for every 2 participantsthere was a time when they were together, then we supose that for any n=k participants there was a time that they were together.
Now assume that there wasn't a time when k+1 participants were together
Now let's call them with $p_1,p_2,...,p_k,p_{k+1}$ and w.l.o.g. say
$T(p_1,p_2,...,p_k) < T(p_2,...,p_k,p_{k+1})<T(p_1,p_3,...,p_{k+1}) $ , where T is the time when they have been together so:
in order to happen second one $p_1$ should go home before and to happen third one after second one $p_1$ should be there but he went home so contradiction.
I don't have any clue about b and c

Answer 1

1. First person to come to the party has to wait for last person to have drink with him or her. So should every other person. So when the last person entered the party, n-1 people were there. Because if somebody left, they couldn't have had drink with last person. Here I assume that if someone left, they won't return to the party.

2. The time of party starts when first person enters and ends when last person has had drink with every other person (assuming n-1 people had drink among themselves before the last person came to finish the party asap).

3. The party will never end and the room needs to be really large to accomodate infinite people.

Answer 2

If nobody go home and come back, then the time when the first person leaves, everybody is at the party.