# 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???

Remark: If you restrict your attention  to sets in these families containing an element m and delete m from all of them, you get another example of such families of sets, possibly with smaller value of t. (Those families which do not include any set containing m will vanish.)

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.

# Francisco Santos Disproves the Hirsch Conjecture

A title and an abstract for the conference “100 Years in Seattle: the mathematics of Klee and Grünbaum” drew a special attention:

Title: “A counter-example to the Hirsch conjecture”

Author: Francisco Santos, Universidad de Cantabria

I have been in Seattle only once, in January 2002, when I visited to give a colloquium talk at UW. Although Victor Klee was already retired–he was 76 years old–he came to the Department of Mathematics to talk to me. We had a nice conversation during which he asked “Why don’t you try to disprove the Hirsch Conjecture”?

I have later found out that he asked the same question to many people, including all his students, but the question and the way it was posed made me feel special at that time. This talk is the answer to that question. I will describe the construction of a 43-dimensional polytope with 86 facets and diameter bigger than 43. The proof is based on a generalization of the $d$-step theorem of Klee and Walkup.

Great news! Congratulations Paco!

Hirsch’s conjecture (1957) states that the  graph of a d-dimensional polytope with n facets  has diameter no more than nd. That is, any two vertices of the polytope may be connected to each other by a path of at most nd edges.

We had quite a few posts regarding the Hirsch conjecture and related problems.

# The Polynomial Hirsch Conjecture: Discussion Thread, Continued

Here is a  link for the just-posted paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle,  Sasha Razborov, and Thomas Rothvoss.

And here is a link to the paper  by Sandeep Koranne and Anand Kulkarni “The d-step Conjecture is Almost true”  – most of the discussion so far was in this direction.

We had a long and interesting discussion regarding the Hirsch conjecture and I would like to continue the discussion here.

The way I regard the open collaborative efforts is as an open collective attempt to discuss and make progress on the problem (and to raise more problems), and also as a way to assist people who think or work (or will think or will work) on these problems on their own.

Most of the discussion in the previous thread was not about the various problems suggested there but rather was about trying to prove the Hirsch Conjecture precisely! In particular, the approach of Sandeep Koranne and Anand Kulkarni which attempts to prove the conjecture using “flips” (closely related to Pachner moves, or bistaller operations) was extensively discussed.  Here is the link to another paper by Koranne and Kulkarni “Combinatorial Polytope Enumeration“. There is certainly more to be understood regarding flips, Pachner moves, the diameter, and related notions. For example, I was curious about for which Pachner moves  “vertex decomposibility” (a strong form of shellability known to imply the Hirsch bound) is preserved. We also briefly discussed metric aspects of the Hirsch conjecture and random polytopes.

For general background: Here is a  chapter that I wrote about graphs, skeleta and paths of polytopes. Some papers on polytopes on Gunter Ziegler’s homepage  describe very interesting and possibly relevant current research in this area. Here is a  link to Eddie Kim and Francisco Santos’s survey article on the Hirsch Conjecture.

Here is a link from the open problem garden to the continuous analog of the Hirsch conjecture proposed by Antoine Deza, Tamas Terlaky, and  Yuriy Zinchenko.