Probability and Statistics Seminar: Balancing Posets and the 1/3-2/3 Conjecture

Name: Probability and Statistics Seminar: Balancing Posets and the 1/3-2/3 Conjecture
Start: 2026-11-04T23:30:00+00:00
End: 2026-11-05T00:30:00+00:00
Location: Kaprielian Hall (KAP)

Wednesday 4 November 2026, 15:30

Venue time (Los Angeles)

Kaprielian Hall (KAP) · Directions

Wednesday 4 November 2026, 15:30Venue time (Los Angeles)

Wednesday 4 November 202615:30 · Venue time (Los Angeles)

Kaprielian Hall (KAP)Directions

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Max Aires, USC

Title: Balancing Posets and the 1/3-2/3 Conjecture

Abstract: A linear extension of P is a total ordering compatible with the poset relations. Let p(x<y) be the probability that x precedes y in a uniformly random linear extension, and let δ(x,y)=min(p(x<y),p(y<x)) and δ(P) be the maximum value of δ(x,y) over all x,y in P. The famous "1/3-2/3" Conjecture, first made over 50 years ago, posits that δ(P)≥1/3 for all posets with at least 2 linear extensions. For three decades, the best progress on this conjecture was a lower bound of (5-√5)/10, which is tight in the case of infinite posets. We discuss a recent improvement beyond this barrier by an (extremely tiny) positive absolute constant. We also discuss the successful resolution of the related "Kahn-Saks Conjecture" which concerns δ(P) in posets of large width. Both these results combine essentially all existing techniques for finding balanced pairs in posets, and were found by a mix of human labor and AI.