# 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 $d$-dimensional polytopes with $n$ facets. Our research so far was devoted to an abstract combinatorial setting. We studied an appealing conjecture by Nicolai Hahnle and considered an even more general abstraction proposed by Yury Volvovskiy. Comments towards this abstract conjecture are most welcome!

Here, I would like to mention a topological approach which follows a result that was discovered independently by Tamon Stephen and Hugh Thomas in their paper An Euler characteristic proof that 4-prismatoids have width at most 4,
and by Paco Santos in his paper Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids. This post is based on a discussion with Paco Santos at Oberwolfach.

## Two maps on a two dimensional Sphere

Theorem: Given a red map and a blue map drawn in general position on $S^2$ there is an intersection point of two edges of different colors which is adjacent (in the refined map) to a red vertex and to a blue vertex.

Blue and black maps

There are two proofs for the theorem. The proof by Stephen and Thomas uses an Euler characteristic argument. The proof by Santos applies a connectivity argument. Both papers are short and elegant. Both papers point out that the result does not hold for maps on a torus.

Santos’ counterexample to the Hirsch conjecture is based on showing that a direct extension of this result to maps in three dimensions fails. (Even for two maps coming from fans based on polytopes.) Of course, first Paco found his counterexample and then the two-map theorem was found in response to his question  of whether one can find in dimension four counterexamples of the kind he presented in dimension five.

The theorem by Santos, Stephen, and Thomas is very elegant. The proofs are simple but far from obvious and it seems to me that the result will find interesting applications. Its elegance and depth reminded me of Anton Klyachko’s car crash theorem.

## A topological question in high dimensions

Now we are ready to present a higher-dimensional analog:

Tentative Conjecture: Let $M_1$ be a red map and let  $M_2$ be a blue map drawn in general position on $S^{n}$, and let $M$ be their common refinement.  There is a vertex $w$ of $M$ a blue vertex $v \in M_1$, a red vertex $u \in M_2$ and two faces $F,~G \in M$ such that 1) $v,w \in F$, 2) $w,u \in G$, and 3) $\dim F + \dim G =n$.

A simple (but perhaps not the most general) setting in which to ask this question is with regard to the red and blue maps  coming from red and blue polyhedral fans associated to red and blue convex polytopes. The common refinement will be the fan obtained by taking all intersections of cones, one from the first fan and one from the second.

(Perhaps when $n=2k$ we can even guarantee that $\dim F=\dim G=k$.)

## Why the tentative conjecture implies that the diameter is polynomial

An affirmative answer to this conjecture will lead to a bound of the form $dn$ for the graph of $d$-polytopes with $n$ facets.

Here is why:

– It is known that the diameter of every polytope with $n$ facets and dimension $d$ is bounded above by the “length” of a Dantzig figure with $2n-2d$ facets and $n-d$ vertices.

Here a Dantzig figure is a simple polytope of dimension $D$ with $2D$ facets and two complementary vertices. (i.e., two vertices such that each vertex lies in half of the facets, and they do not belong to any common facet).

The length of the Dantzig figure is the graph distance between these two vertices. This is the classical “d-step theorem” of Klee and Walkup, 1967.

– The length of a Dantzig figure of dimension $d$ is the same as the minimum distance between blue and red vertices in a pair of two maps in the $(d-2)$-sphere, with $d$ cells each.

– Our tentative conjecture implies, by dimension on $d$, that the minimum distance between blue and red vertices in a pair of maps in the $d$-sphere and with $n$ cells is bounded above by (essentially) $nd$. ($n$ cells means “cells of the blue map plus cells of the red map”, not “cells of the common refinement”).

## The abstract setting and other approaches

More comments, ideas, and updates on the abstract setting are of course very welcome Also very welcome are other approaches to the polynomial Hirsch conjecture, and discussion of related problems.

An example showing that the theorem fail for blue and red maps on a torus.

# Polynomial Hirsch Conjecture 5: Abstractions and Counterexamples.

This is the 5th research thread of polymath3 studying the polynomial Hirsch conjecture. As you may remember, we are mainly interested in an abstract form of the problem about families of sets. (And a related version about families of multisets.)

The 4th research thread was, in my opinion, fruitful. An interesting further abstraction was offered and for this abstraction a counterexample was found. I will review these developments below.

There are several reasons why the positive direction is more tempting than the negative one. (And as usual, it does not make much of a difference which direction you study. The practices for trying to prove a statement and trying to disprove it are quite similar.) But perhaps we should try to make also some more pointed attempts towards counterexamples?

Over the years, I devoted much effort including a few desperate attempts to try to come up with counterexamples. (For a slightly less abstract version than that of EHRR.) I tried to base one on the Towers of Hanoi game. One can translate the positions of the game into a graph labelled by subsets. But the diameter is exponential! So maybe there is a way to change the “ground set”? I did not find any. I even tried to look at games (in game stores!) where the player is required to move from one position to another to see if this leads to an interesting abstract example. These were, while romantic, very long shots.

Two more things: First, I enjoyed meeting in Lausanne for the first time Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss. (EHR of EHRR.) Second, Oliver Friedmann, Thomas Dueholm Hansen, and Uri Zwick proved (mildly) subexponential lower bounds for certain randomized pivot steps for the simplex algorithm. We discussed it in this post.  The underlying polytopes in their examples are combinatorial cubes. So this has no direct bearing on our problem. (But it is interesting to see if geometric or abstract examples coming from more general games of the type they consider may be relevant.)

So let me summarize PHC4 excitements and, as usual, if I missed something please add it.

# Polymath3: Polynomial Hirsch Conjecture 4

So where are we? I guess we are trying all sorts of things, and perhaps we should try even more things. I find it very difficult to choose the more promising ideas, directions and comments as Tim Gowers and Terry Tao did so effectively in Polymath 1,4 and 5.  Maybe this part of the moderator duty can also be outsourced. If you want to point out an idea that you find promising, even if it is your own idea, please, please do.

This post has three parts. 1) Around Nicolai’s conjecture; 1) Improving the upper bounds based on the original method; 3) How to find super-polynomial constructions? Continue reading

# Polymath3 : Polynomial Hirsch Conjecture 3

Here is the third research thread for the polynomial Hirsch conjecture.  I hope that people will feel as comfortable as possible to offer ideas about the problem we discuss. Even more important, to think about the problem either in the directions suggested by others or on their own. Participants who follow the project and think about the issues without adding remarks are valuable.

The combinatorial problem is simple to state and also everything that we know about it is rather simple. At this stage joining the project should be easy.

Let me try to describe (without attemting to be complete) one main direction that we discuss. This direction started with the very first comment we had by Nicolai.

Please do not hesitate to repeat an idea raised by yourself or by other if you think it can be useful.

Let $f^*(d,n)$ be the largest number of disjoint families $F_1, F_2, ..., F_t$ of degree d monomials in the variables $x_1,\dots,x_n$ such that

(*) for i < j < k, whenever $m_1 \in F_i$ and $m_2 \in F_k$, then there exists a monomial $u \in F_j$ such that $gcd(m_1, m_2) | u$.

### Nicolai’s conjecture:

$f^*(d,n)=d(n-1)+1$.

The example that supports this conjecture consists of families with a single monomial in every family.

The monomials are

$x_1^d,$

$x_1^{d-1}x_2$,

$\dots$,

$x_2^d$,

$x_2^{d-1}x_3$,

$\dots$,

$x_n^d$.

There are other examples that achieve the same bound. The bound can be achieved by families whose union include all monomials, and for such families the conjecture is correct.

### The case d=3.

An upper bound by EHRR (that can be extended to monomials) following works of Barnette and Larman on polytopes is $f^*(d,n) \le 2^{d-1}n$. For degree 3 monomials we have a gap

$3n-2\le f^*(3,n) \le 4n-1$.

It may be the case that understanding the situation for $d=3$ is the key for the whole problem.

There is another example achieving the lower bound that Terry found

$F_i := \{ \{a,b,c\}: a+b+c = i+2 \}$ $i=1,2,\dots 3n-2$

### Various approaches to the conjecture

Several approaches to the cojecture were proposed. Using clever reccurence relations, finding useful ordering, applying the method of compression, and algebraic methods. In a series of remarks Tim is trying to prove Nicolai’s conjecture. An encouraging sign is that both examples of Nicolai, Klas, and Terry come up naturally. One way to help the project at this stage would be to try to enter Tim’s mind and find ways to help him “push the car”. In any case, if Nicolai’s conjecture is correct I see no reason why it shouldn’t have a simple proof (of course we will be happy with long proofs as well).

### Constructions

Something that is also on the back of our minds is the idea to find examples that are inspired from the upper bound proofs. We do not know yet what direction is going to prevail so it is useful to remember that every proof of a weaker result and every difficulty in attempts to proof the hoped-for result can give some ideas for disproving what we are trying to prove.

Some preliminary attempts were made to examine what are the properties of examples for d=3 which will come close to the 4n bound. It may also be the case that counterexamples to Nicolai’s conjecture can be found for rather small values of n and d.

Two polls:

# Polymath 3: The Polynomial Hirsch Conjecture 2

Here we start the second research thread about the polynomial Hirsch conjecture.  I hope that people will feel as comfortable as possible to offer ideas about the problem. The combinatorial problem looks simple and also everything that we know about it is rather simple: At this stage joining the project should be very easy. If you have an idea (and certainly a question or a request,) please don’t feel necessary to read all earlier comments to see if it is already there.

In the first post we described the combinatorial problem: Finding the largest possible number f(n) of disjoint families of subsets from an n-element set which satisfy a certain simple property (*).We denote by f(d,n) the largest possible number of families satisfying (*) of d-subsets from {1,2,…,n}.

The two principle questions we ask are:

Can the upper bounds be improved?

and

Can the lower bounds be improved?

What are the places that the upper bound argument is wasteful and how can we improve it? Can randomness help for constructions? How does a family for which the upper bound argument is rather sharp will look like?

We are also interested in the situation for small values of n and for small values of d. In particular, what is f(3,n)? Extending the problem to multisets (or monomials) instead of sets may be fruitful since there is a proposed suggestion for an answer.

# Polymath 3: Polynomial Hirsch Conjecture

I would like to start here a research thread of the long-promised Polymath3 on the polynomial Hirsch conjecture.

I propose to try to solve the following purely combinatorial problem.

Consider t disjoint families of subsets of {1,2,…,n}, $F_1, F_2, ..., F_t$.

Suppose that

(*) For every $i, and every $S \in F_i$ and $T \in F_k$, there is $R\in F_j$ which contains $S\cap T$.

The basic question is: How large can t  be???

(When we say that the families are disjoint we mean that there is no set that belongs to two families. The sets in a single family need not be disjoint.)

In a recent post I showed the very simple argument for an upper bound $n^{\log n+1}$. The major question is if there is a polynomial upper bound. I will repeat the argument below the dividing line and explain the connections between a few versions.

A polynomial upper bound for $f(n)$ will imply a polynomial (in $n$) upper bound for the diameter of graphs of polytopes with $n$ facets. So the task we face is either to prove such a polynomial upper bound or give an example where $t$ is superpolynomial.

The abstract setting is taken from the paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle,  Sasha Razborov, and Thomas Rothvoss. They gave an example that $f(n)$ can be quadratic.

Remark: The comments for this post will serve both the research thread and for discussions. I suggested to concentrate on a rather focused problem but other directions/suggestions are welcome as well.

# The Polynomial Hirsch Conjecture: The Crux of the Matter.

Consider t disjoint families of subsets of {1,2,…,n}, $F_1, F_2, ..., F_t$.

Suppose that (*) For every $i, and every $S \in F_i$ and $T \in F_k$, there is $R\in F_j$  which contains $S\cap T$

The basic question is: How large can t  be???

Theorem: $f(n)\le f(n-1)+2f(n/2)$.
Proof: Consider the largest s so that the union of all sets in $F_1,...,F_s$ is at most n/2.   Clearly, $s \le f(n/2)$.
Consider the largest r so that the union of all sets in $F_{t-r+1},...,F_t$ is at most n/2.   Clearly, $r\le f(n/2)$.
Now, by the definition of s and r, there is an element m shared by a set in the first s+1 families and a set in the last r+1 families. Therefore (by (*)), when we restrict our attention to the sets containing ‘m’ the families $F_{s+1},...,F_{t-r}$ all survive. We get that $t-r-s\le f(n-1)$. Q.E.D.