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# Category Archives: Open problems

## 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

## How Large can a Spherical Set Without Two Orthogonal Vectors Be?

The problem Witsenhausen’s Problem (1974): Let be a measurable subset of the -dimensional sphere . Suppose that does not contain two orthogonal vectors. How large can the -dimensional volume of be? A Conjecture Conjecture: The maximum volume is attained … Continue reading

Posted in Open problems
4 Comments

## The Cap-Set Problem and Frankl-Rodl Theorem (C)

Update: This is a third of three posts (part I, part II) proposing some extensions of the cap set problem and some connections with the Frankl Rodl theorem. Here is a post presenting the problem on Terry Tao’s blog (March 2007). Here … Continue reading

Posted in Combinatorics, Open problems
Tagged Cap sets, Frankl-Rodl theorem, polymath1
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## Around the Cap-Set problem (B)

Part B: Finding special cap sets This is a second part in a 3-part series about variations on the cap set problem that I studied with Roy Meshulam. (The first post is here.) I will use here a different notation than in part … Continue reading

## A Problem on Planar Percolation

Conjecture (Gady Kozma): Prove that the critical probability for planar percolation on a Cayley graph of the group is always an algebraic number. Gady mentioned this conjecture in his talk here about percolation on infinite Cayley graphs. (Update April 30: Today Gady mentioned … Continue reading

## (Eran Nevo) The g-Conjecture II: The Commutative Algebra Connection

Richard Stanley This post is authored by Eran Nevo. (It is the second in a series of five posts.) The g-conjecture: the commutative algebra connection Let be a triangulation of a -dimensional sphere. Stanley’s idea was to associate with a ring … Continue reading

## How the g-Conjecture Came About

This post complements Eran Nevo’s first post on the -conjecture 1) Euler’s theorem Euler Euler’s famous formula for the numbers of vertices, edges and faces of a polytope in space is the starting point of many mathematical stories. (Descartes came close … Continue reading

## (Eran Nevo) The g-Conjecture I

This post is authored by Eran Nevo. (It is the first in a series of five posts.) Peter McMullen The g-conjecture What are the possible face numbers of triangulations of spheres? There is only one zero-dimensional sphere and it consists … Continue reading

Posted in Combinatorics, Convex polytopes, Guest blogger, Open problems
Tagged face rings, g-conjecture, Polytopes
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## An Open Discussion and Polls: Around Roth’s Theorem

Suppose that is a subset of of maximum cardinality not containing an arithmetic progression of length 3. Let . How does behave? We do not really know. Will it help talking about it? Can we somehow look beyond the horizon and try to guess what … Continue reading

Posted in Combinatorics, Open discussion, Open problems
Tagged Cap sets, polymath1, Roth's theorem, Szemeredi's theorem
25 Comments

## Rosenfeld’s Odd-Distance Problem

Moshe Rosenfeld’s odd-distance problem: Let G be the graph whose vertices are points in the plane and two vertices form an edge if their distance is an odd integer. Is the chromatic number of this graph finite?

Posted in Combinatorics, Open problems
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