Category Archives: Open problems

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

Posted in Combinatorics, Convex polytopes, Open problems | Tagged , , , | 4 Comments

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

Posted in Combinatorics, Open discussion, Open problems | Tagged , , , | 24 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 | 1 Comment

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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Telling a Simple Polytope From its Graph

Peter Mani  (a photograph by Emo Welzl) Simple polytopes, puzzles   Micha A. Perles conjectured in the ’70s that the graph of a simple -polytope determines the entire combinatorial structure of the polytope. This conjecture was proved in 1987 by Blind … Continue reading

Posted in Convex polytopes, Open problems | Tagged , , | 4 Comments

Lovasz’s Two Families Theorem

Laci and Kati This is the first of a few posts which are spin-offs 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 , , | 4 Comments