Question

Answer the following questions-
1) Does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?

2) If {k+1}k+1 runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance {1/(k+1)}1/(k+1) from each other runner) at some time?

Answer 1

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