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 Friendship and Sesame, Maryam and Marina, Israel and Iran
 Elchanan Mossel’s Amazing Dice Paradox (your answers to TYI 30)
 TYI 30: Expected number of Dice throws
 Test your intuition 29: Diameter of various random trees
 Micha Perles’ Geometric Proof of the ErdosSos Conjecture for Caterpillars
 Touching Simplices and Polytopes: Perles’ argument
 Where were we?
 Call for nominations for the Ostrowski Prize 2017
 Problems for Imre Bárány’s Birthday!
Top Posts & Pages
 Elchanan Mossel's Amazing Dice Paradox (your answers to TYI 30)
 Friendship and Sesame, Maryam and Marina, Israel and Iran
 TYI 30: Expected number of Dice throws
 Test your intuition 29: Diameter of various random trees
 Answer: Lord Kelvin, The Age of the Earth, and the Age of the Sun
 A Breakthrough by Maryna Viazovska Leading to the Long Awaited Solutions for the Densest Packing Problem in Dimensions 8 and 24
 Test Your Intuition (17): What does it Take to Win TicTacToe
 Polymath 10 Emergency Post 5: The ErdosSzemeredi Sunflower Conjecture is Now Proven.
 Touching Simplices and Polytopes: Perles' argument
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Category Archives: Open problems
Polymath 3: Polynomial Hirsch Conjecture
I would like to start here a research thread of the longpromised Polymath3 on the polynomial Hirsch conjecture. I propose to try to solve the following purely combinatorial problem. Consider t disjoint families of subsets of {1,2,…,n}, . Suppose that … Continue reading
Posted in Convex polytopes, Open discussion, Open problems, Polymath3
Tagged Hirsch conjecture, Polymath3
120 Comments
The Polynomial Hirsch Conjecture: The Crux of the Matter.
Consider t disjoint families of subsets of {1,2,…,n}, . Suppose that (*) For every , and every and , there is which contains . The basic question is: How large can t be??? Let’s call the answer f(n). … Continue reading
Posted in Combinatorics, Convex polytopes, Open problems, Polymath3
5 Comments
Francisco Santos Disproves the Hirsch Conjecture
A title and an abstract for the conference “100 Years in Seattle: the mathematics of Klee and Grünbaum” drew a special attention: Title: “A counterexample to the Hirsch conjecture” Author: Francisco Santos, Universidad de Cantabria Abstract: I have been in … Continue reading
Posted in Convex polytopes, Open problems, Polymath3
36 Comments
The Polynomial Hirsch Conjecture: Discussion Thread, Continued
Here is a link for the justposted paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle, Sasha Razborov, and Thomas Rothvoss. And here is a link to the paper by Sandeep Koranne and Anand Kulkarni “The dstep Conjecture is Almost true” – … Continue reading
Posted in Convex polytopes, Open discussion, Open problems
Tagged Convex polytopes, Hirsch conjecture
16 Comments
(Eran Nevo) The gConjecture III: Algebraic Shifting
This is the third in a series of posts by Eran Nevo on the gconjecture. Eran’s first post was devoted to the combinatorics of the gconjecture and was followed by a further post by me on the origin of the gconjecture. … Continue reading
Posted in Combinatorics, Convex polytopes, Guest blogger, Open problems
Tagged gconjecture, Shifting
4 Comments
The Polynomial Hirsch Conjecture: Discussion Thread
This post is devoted to the polymathproposal about the polynomial Hirsch conjecture. My intention is to start here a discussion thread on the problem and related problems. (Perhaps identifying further interesting related problems and research directions.) Earlier posts are: The polynomial Hirsch … Continue reading
Posted in Convex polytopes, Open discussion, Open problems
Tagged Hirsch conjecture, Polytopes
115 Comments
Polymath4 – Finding Primes Deterministically – is On Its Way
After two long and interesting discussion threads polymath4, devoted to finding deterministically large prime numbers, is on its way on the polymath blog.
The Polynomial Hirsch Conjecture – How to Improve the Upper Bounds.
I can see three main avenues toward making progress on the Polynomial Hirsch conjecture. One direction is trying to improve the upper bounds, for example, by looking at the current proof and trying to see if it is wasteful and if so where … Continue reading
Posted in Convex polytopes, Open discussion, Open problems
Tagged Discussion, Hirsch conjecture
14 Comments
The Polynomial Hirsch Conjecture, a Proposal for Polymath3 (Cont.)
The Abstract Polynomial Hirsch Conjecture A convex polytope is the convex hull of a finite set of points in a real vector space. A polytope can be described as the intersection of a finite number of closed halfspaces. Polytopes have … Continue reading
Posted in Convex polytopes, Open discussion, Open problems
Tagged Hirsch conjecture, Polymath proposals
5 Comments
The Polynomial Hirsch Conjecture: A proposal for Polymath3
This post is continued here. Eddie Kim and Francisco Santos have just uploaded a survey article on the Hirsch Conjecture. The Hirsch conjecture: The graph of a dpolytope with n vertices facets has diameter at most nd. We devoted several … Continue reading