# Everybody present

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

• For (c): Consider what happens if person $i$ is at the party for time $t$ precisely when $0<t\leq\frac{1}{i}$. So person 1 is at the party during the interval $(0,1]$, person 2 during the interval $(0,1/2]$, etc. – Danny Rorabaugh Oct 12 '16 at 17:31
• You could compare the time the first person left (or the time the first person to stop drinking stopped drinking) with the time the last person arrived (or the time the last person to start drinking started drinking). You hope nobody took a break from the party. You might have to deal with suprema and infima if there are an infinite number of people as there might be not a first and last, but people may not be present at these points – Henry Oct 12 '16 at 17:31