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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
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 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
 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 29: Diameter of various random trees
 The Race to Quantum Technologies and Quantum Computers (Useful Links)
 Touching Simplices and Polytopes: Perles' argument
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Tag Archives: Topological combinatorics
Polymath10, Post 2: Homological Approach
We launched polymath10 a week ago and it is time for the second post. In this post I will remind the readers what the ErdosRado Conjecture and the ErdosRado theorem are, briefly mention some points made in the previous post and in … Continue reading
Polymath3 (PHC6): The Polynomial Hirsch Conjecture – A Topological Approach
This is a new polymath3 research thread. Our aim is to tackle the polynomial Hirsch conjecture which asserts that there is a polynomial upper bound for the diameter of graphs of dimensional polytopes with facets. Our research so far was … Continue reading
Posted in Convex polytopes, Geometry, Polymath3
Tagged Hirsch conjecture, Polymath3, Topological combinatorics
37 Comments
A Beautiful Garden of Hypertrees
We had a series of posts (1,2,3,4) “from Helly to Cayley” on weighted enumeration of Qacyclic simplicial complexes. The simplest case beyond Cayley’s theorem were Qacyclic complexes with vertices, edges, and triangles. One example is the sixvertex triangulation of the … Continue reading
Posted in Combinatorics
Tagged Mishael Rosenthal, Nati Linial, Roy Meshulam, Topological combinatorics, Trees
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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
5 Comments
Helly’s Theorem, “Hypertrees”, and Strange Enumeration I
1. Helly’s theorem and Cayley’s formula Helly’s theorem asserts: For a family of n convex sets in , n > d, if every d+1 sets in the family have a point in common then all members in the family have a point in common. … Continue reading
Posted in Combinatorics, Convexity
Tagged Cayley theorem, Helly Theorem, Simplicial complexes, Topological combinatorics, Trees
8 Comments
A Small Debt Regarding Turan’s Problem
Turan’s problem asks for the minimum number of triangles on n vertices so that every 4 vertices span a triangle. (Or equivalently, for the maximum number of triangles on n vertices without a “tetrahedron”, namely without having four triangles on … Continue reading