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Richard P. Stanley Seminar in Combinatorics: Tight Bound and Structural Theorem for Joints

SEMINARS: HARVARD-MIT COMBINATORICS

When: October 20, 2023
3:00 pm - 4:00 pm
Where: MIT, Room 2-139
Speaker: Ting-Wei Chao - CMU | Hung-Hsun Hans Yu - Princeton

The joints problem asks to determine the maximum number of joints N lines can form, where a joint in a d-dimensional space is a point on d lines in linearly independent directions. Recently, we determined the maximum exactly for k choose d-1 lines in d-dimensional space, namely k choose d. What is more important is that we are able to prove a structural result determining all optimal configurations, and this is the first success of the polynomial method in this direction. It turns out that our result implies a conjecture of Bollobás and Eccles as an immediate corollary regarding a generalization of the Kruskal–Katona theorem. In this talk, we will talk about the connection to that conjecture and also give a high-level overview of the key ideas.

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For more info, see https://math.mit.edu/combin/