# Category Archives: Open problems

## Some Problems

Four posts ago I wrote about three recent breakthroughs in combinatorics and in the following post I would like to mention some problems that I posed over the years that are loosely related to these advances. Rank of incidence matrices … Continue reading

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## A Nice Example Related to the Frankl Conjecture

Updates: 1. Peter Frankl brought to my attention that the very same example appeared in a paper by Dynkin and Frankl “Extremal sets of subsets satisfying conditions induced by a graph“. 2. Sam Hopkins gave a lovely reference to Ravi … Continue reading

## Amazing: Justin Gilmer gave a constant lower bound for the union-closed sets conjecture

Frankl’s conjecture (aka the union closed sets conjecture) asserts that if is a family of subsets of [n] (=: ) which is closed under union then there is an element such that Justin Gilmer just proved an amazing weaker form … Continue reading

Posted in Combinatorics, Open problems | | 22 Comments

## Open problem session of HUJI-COMBSEM: Problem #5, Gil Kalai – the 3ᵈ problem

This post continues to describe problems presented at our open problems session back in November 2020. Here is the first post in the series.  Today’s problem was presented by me, and it was an old 1989 conjecture of mine. A … Continue reading

## Open problem session of HUJI-COMBSEM: Problem #1, Nati Linial – Turan type theorems for simplicial complexes.

On November, 2020  we had a very nice open problem session in our weekly combinatorics seminar at HUJI.  So I thought to have a series of posts to describe you the problems presented there.  This is the first post in … Continue reading

Posted in Combinatorics, Geometry, Open problems | Tagged | 3 Comments

## Péter Pál Pach and Richárd Palincza: a Glimpse Beyond the Horizon

Prologue Consider the following problems: P3: What is the maximum density of a set A in without a 3-term AP? (AP=arithmetic progression.) This is the celebrated Cap Set problem and we reported here in 2016 the breakthrough results by … Continue reading

Posted in Combinatorics, Geometry, Number theory, Open problems | | 9 Comments

## Kelman, Kindler, Lifshitz, Minzer, and Safra: Towards the Entropy-Influence Conjecture

Let me briefly report on a remarkable new paper by Esty Kelman, Guy Kindler, Noam Lifshitz, Dor Minzer, and Muli Safra, Revisiting Bourgain-Kalai and Fourier Entropies. The paper describes substantial progress towards the Entropy-Influence conjecture, posed by Ehud Friedgut and … Continue reading

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## Ringel Conjecture, Solved! Congratulations to Richard Montgomery, Alexey Pokrovskiy, and Benny Sudakov

Ringel’s conjecture solved (for sufficiently large n) A couple weeks ago and a few days after I heard an excellent lecture about it by Alexey Pokrovskiy in Oberwolfach, the paper A proof of Ringel’s Conjecture by Richard Montgomery, Alexey Pokrovskiy, … Continue reading

Posted in Combinatorics, Open problems, Updates | | 5 Comments

## Test your intuition 43: Distribution According to Areas in Top Departments.

In the community of mamathetitians in a certain country there are mamathetitians in two areas: Anabra (fraction p of the mamathetitians) and Algasis (fraction 1-p of  mamathetitians.) There are ten universities with 50 faculty members in each mamathetics department … Continue reading

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## Gérard Cornuéjols’s baker’s eighteen 5000 dollars conjectures

Gérard Cornuéjols Gérard Cornuéjols‘s beautiful (and freely available) book from 2000 Optimization: Packing and Covering is about an important area of combinatorics which is lovely described in the preface to the book The integer programming models known as set packing … Continue reading