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Category Archives: Open problems
Updates and Plans V: From Boise to Tel Aviv, Ceasefire, My 70th Birthday, Nostalgia, Problems, Outrageous Conjectures, Quantum, and AI
This is the fifth post of this type (I (2008); II(2011); III(2015); IV(2024)). Between Boise and Tel Aviv During the summer we spent two months in the lovely city of Boise, Idaho. We stayed with my son Hagai and his husband Felix, … Continue reading
Posted in Combinatorics, Computer Science and Optimization, Open problems, personal, Updates
Tagged Benjamin Weiss
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Richard Montgomery and Lisa Sauermann Present Major Progress on Rota’s Basis Conjecture
Richard Montgomery and Lisa Sauermann: Asymptotically-tight packing and covering with transversal bases in Rota’s basis conjecture Abstract: In 1989, Rota conjectured that, given any bases of a vector space of dimension , or more generally a matroid of rank , … Continue reading
Posted in Combinatorics, Open problems
Tagged Gian Carlo Rota, Lisa Sauermann, Polymath12, Richard Montgomery
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Viterbo’s conjecture was refuted by Pazit Haim-Kislev and Yaron Ostrover
Viterbo conjecture – refuted Claude Viterbo’s 2000 volume-capacity conjecture asserts that the Euclidean (even dimensional) ball maximizes (every) symplectic capacity among convex bodies of the same volume. In the recent paper A Counterexample to Viterbo’s Conjecture, Pazit Haim-Kislev and Yaron … Continue reading
Andrii Arman, Andriy Bondarenko, Fedor Nazarov, Andriy Prymak, and Danylo Radchenko Constructed Small Volume Bodies of Constant Width
From left to right: Andrii Arman, Andriy Bondarenko and Danylo Radchenko, Fedor Nazarov, and Andriy Primak. The -dimensional unit Euclidean ball has width 2 in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction … Continue reading
Posted in Convexity, Open problems, Updates
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What is the maximum number of Tverberg’s partitions?
The problem presented in this post was discussed in my recent lecture “New types of order types” in the workshop on discrete convexity and geometry in Budapest, a few weeks ago. The lecture described various results and questions including the … Continue reading
Some Problems
Four posts ago I wrote about three recent breakthroughs in combinatorics and in the following post I would like to mention some problems that I posed over the years that are loosely related to these advances. Rank of incidence matrices … Continue reading
A Nice Example Related to the Frankl Conjecture
Updates: 1. Peter Frankl brought to my attention that the very same example appeared in a paper by Dynkin and Frankl “Extremal sets of subsets satisfying conditions induced by a graph“. 2. Sam Hopkins gave a lovely reference to Ravi … Continue reading
Amazing: Justin Gilmer gave a constant lower bound for the union-closed sets conjecture
Frankl’s conjecture (aka the union closed sets conjecture) asserts that if is a family of subsets of [n] (=: ) which is closed under union then there is an element such that Justin Gilmer just proved an amazing weaker form … Continue reading
Open problem session of HUJI-COMBSEM: Problem #5, Gil Kalai – the 3ᵈ problem
This post continues to describe problems presented at our open problems session back in November 2020. Here is the first post in the series. Today’s problem was presented by me, and it was an old 1989 conjecture of mine. A … Continue reading
Open problem session of HUJI-COMBSEM: Problem #1, Nati Linial – Turan type theorems for simplicial complexes.
On November, 2020 we had a very nice open problem session in our weekly combinatorics seminar at HUJI. So I thought to have a series of posts to describe you the problems presented there. This is the first post in … Continue reading
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