Category Archives: Open problems

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

The problem Problem: 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 by two … Continue reading

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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

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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

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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

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(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

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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

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(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

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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

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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?  

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Frankl-Rodl’s Theorem and Variations on the Cap Set Problem: A Recent Research Project with Roy Meshulam (A)

Voita Rodl I would like to tell you about a research project in progress with Roy Meshulam. (We started it in the summer, but then moved to other things;  so far there are interesting insights, and perhaps problems, but not substantial … Continue reading

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