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 Postoctoral Positions with Karim and Other Announcements!
 Jirka
 AviFest, AviStories and Amazing Cash Prizes.
 Polymath 10 post 6: The ErdosRado sunflower conjecture, and the Turan (4,3) problem: homological approaches.
 Polymath 10 Emergency Post 5: The ErdosSzemeredi Sunflower Conjecture is Now Proven.
 Mind Boggling: Following the work of Croot, Lev, and Pach, Jordan Ellenberg settled the cap set problem!
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 The Erdős Szekeres polygon problem – Solved asymptotically by Andrew Suk.
 The Quantum Computer Puzzle @ Notices of the AMS
Top Posts & Pages
 Postoctoral Positions with Karim and Other Announcements!
 The Erdős Szekeres polygon problem  Solved asymptotically by Andrew Suk.
 Polymath 10 Emergency Post 5: The ErdosSzemeredi Sunflower Conjecture is Now Proven.
 A Breakthrough by Maryna Viazovska Leading to the Long Awaited Solutions for the Densest Packing Problem in Dimensions 8 and 24
 Emmanuel Abbe: Erdal Arıkan's Polar Codes
 Jirka
 A Few Mathematical Snapshots from India (ICM2010)
 Believing that the Earth is Round When it Matters
 Mind Boggling: Following the work of Croot, Lev, and Pach, Jordan Ellenberg settled the cap set problem!
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Category Archives: Convexity
Raigorodskii’s Theorem: Follow Up on Subsets of the Sphere without a Pair of Orthogonal Vectors
Andrei Raigorodskii (This post follows an email by Aicke Hinrichs.) In a previous post we discussed the following problem: Problem: Let be a measurable subset of the dimensional sphere . Suppose that does not contain two orthogonal vectors. How large … Continue reading
Test Your Intuition (5)
(Not such a set) consider a planar set A with the following property. In every direction, the distance between the two parallel lines that touch A from both sides is the same! Must A be a circle?
Colorful Caratheodory Revisited
Janos Pach wrote me: “I saw that you several times returned to the colored Caratheodory and Helly theorems and related stuff, so I thought that you may be interested in the enclosed paper by Holmsen, Tverberg and me, in … Continue reading
Lovasz’s Two Families Theorem
Laci and Kati This is the first of a few posts which are spinoffs of the extremal combinatorics series, especially of part III. Here we talk about Lovasz’s geometric two families theorem. 1. Lovasz’s two families theorem Here … Continue reading
Posted in Combinatorics, Convexity, Open problems
Tagged exterior algebras, Extremal combinatorics, shellability
5 Comments
Seven Problems Around Tverberg’s Theorem
Imre Barany, Rade Zivaljevic, Helge Tverberg, and Sinisa Vrecica Recall the beautiful theorem of Tverberg: (We devoted two posts (I, II) to its background and proof.) Tverberg Theorem (1965): Let be points in , . Then there is a partition of … Continue reading
Test Your Intuition (2)
Question: Let be the cube in centered at the origin and having dimensional volume equal to one. What is the maximum dimensional volume of when is a hyperplane? Can you guess the behavior of when ? Can you guess the plane which … Continue reading
Sarkaria’s Proof of Tverberg’s Theorem 2
Karanbir Sarkaria 4. Sarkaria’s proof: Tverberg’s theorem (1965): Let be points in , . Then there is a partition of such that . Proof: We can assume that . First suppose that the points belong to the dimensional affine space … Continue reading
Sarkaria’s Proof of Tverberg’s Theorem 1
Helge Tverberg Ladies and gentlemen, this is an excellent time to tell you about the beautiful theorem of Tverberg and the startling proof of Sarkaria to Tverberg’s theorem (two parts). A good place to start is Radon’s theorem. 1. The theorems of Radon, … Continue reading
Helly, Cayley, Hypertrees, and Weighted Enumeration III
This is the third and last part of the journey from a Helly type conjecture of Katchalski and Perles to a Cayley’s type formula for “hypertrees”. (On second thought I decided to divide it into two devoting the second to probabilistic questions.) … Continue reading
Posted in Combinatorics, Convexity, Open problems, Probability
6 Comments
Helly’s Theorem, “Hypertrees”, and Strange Enumeration II: The Formula
In the first part of this post we discussed an appealing conjecture regaring an extension of Cayley’s counting trees formula. The number of ddimensional “hypertrees” should somehow add up to . But it was not clear to us which complexes we want … Continue reading
Posted in Combinatorics, Convexity
Tagged Cayley theorem, Helly type theorems, Topological combinatorics
4 Comments