## Archive for the ‘open questions’ Category

### 2^n \geq n: The graph theory proof?

February 15, 2010

Theorem. For every positive integer n, $2^n \geq n$.

Proof. Consider a tree on n vertices $T = (V, E)$ with root $v_0$. Assign to each edge $\{v, w\}$ an element of the vector space $GF(2)^V$, obtained by setting 1s in the the coordinates corresponding to v and w and 0s elsewhere. I claim that these vectors are linearly independent; for suppose otherwise, and letthe vectors corresponding to $S \subset E$ sum to 0. There is a natural “distance” function on E w/r/t $v_0$; let $e_0 = \{s, t\}$ have maximal distance in S, and suppose WLOG that t is farther than s from $v_0$. Then the coordinate corresponding to t is nonzero for exactly one element of S, and the sum over all elements of S must therefore be nonzero. This is a contradiction. So in particular these |E| = n-1 elements are distinct and nonzero, which means (by Pigeonhole) that there are at most $2^n-1$ of them.

### Full HAP tables for the second 1124 sequence

January 12, 2010

Since it’s not yet on Thomas’ blog, and I need to use the data, I thought I’d make it publicly available. All credit goes to Thomas Sauvaget for the sequence-to-HTML code, and to Polymath for the sequence itself. (Probably someone specific discovered it, but I don’t know who.) Tables below the fold.

### Bleg: What’s the most recent day no one alive was born

January 5, 2010

Inspired by Michael Lugo’s post on reconstructing a person from their DOB, zipcode, and gender.

If you, for whatever reason, ever watch the Today show, you’ll notice that one of the recurring features is the hosts listing the names of some men and women who are turning 100. Becoming a centenarian is a reasonably big accomplishment — in the U.S., it nets you a congratulatory letter from the President, for example. But if you look into it, you’ll notice that you can find someone turning 100 on pretty much any given day. Usually not someone particularly well-known, but certainly someone. (I tried to find someone famous and vaguely math-related who just turned or is turning 100 for this post, but couldn’t; however, the fascinating economist Ronald Coase turned 99 last week.) It’s almost certainly true that on any given day, someone somewhere in the world is in fact celebrating their 100th birthday. But go ten years further, and you find almost no one who lives to 110. Actually, I know of only one supercentenarian, living or not, who is interesting for reasons apart from his longevity — the late Vietoris, the topologist, probably best known as half of the Vietoris-Rips complex and the Mayer-Vietoris sequence. Odds are pretty good that no one alive is turning 110 today, or tomorrow, or (sadly) New Years’ Day.

So… a question is starting to take shape. On every day between December 29, 1909, and today, someone was born who is still living today. But much earlier than that, and the above statement begins to be false. So what’s the most recent day that no one living was born on?

### What’s a “locally determined graph property?”

January 1, 2010

This has nothing to do with the rest of the post, but I’ll put it here so you read it before you get bored. I’d like to thank my readers (all seven of you) for supporting this blog in the first six months or so of its existence, and hope that you’ll stick around (and be joined by hundreds of new readers…) to hear my sporadic ramblings and wild ravings in the next year. Here’s to a happy and successful 2010!

Over at MathOverflow, Gjergji Zaimi asks (in a criminally under-voted-for question): How can we obtain global information from local data in graph theory?  This is something that perhaps everyone working in or around graph theory has asked themselves, in some form, at some point — I know I have. So it’s not surprising that Gjergji’s question has received many different answers with many different interesting things to say.

I originally wanted to write a post trying to “answer” Gjergji’s question as best I could, but quickly realized the futility of that goal — it’s such a broad and deep question that I doubt if anyone could answer it concisely, and I know I couldn’t! So instead I’ll just talk about an $\epsilon$ of the question — what does it even mean, “local data?”

### The importance of choosing the right model

December 13, 2009

I’ve had the idea for this post bouncing around in my head for several months now, but now that I don’t have classes to worry about I can finally get around to writing it. I want to talk about some “pathological examples” in computer science, in particular in complexity and computability theory.

I wanted to post a picture of William H. Mills with this post, in the spirit of Dick Lipton’s blog. Unfortunately I can’t find one! Mills was a student of Emil Artin in the ’40s; he finished his thesis in 1949 and promptly (as far as I can tell) disappeared until the ’70s, when we find some work by a William H. Mills on combinatorial designs. After this, there’s nothing of note until Dr. Mills passed away several years ago at the age of 85.

While his work (assuming it’s his) on combinatorial designs looks interesting, W.H. Mills’ small place in history is assured by a short and unassuming paper from 1947, published when he was still a student. In this paper he showed that there’s a constant, which he called A, such that $[A^{3^n}]$ is prime for all integers $n \geq 1$. Nowadays it’s called Mills’ constant.

### Where do graphs live?

December 5, 2009

This post came out of some thoughts I posted (anonymously, but mostly because I didn’t feel like registering) over at nLab. I don’t think it’s a secret that I’m heavily interested in the relationships between category theory and combinatorics, and more generally the ways in which we can use “structured” algebraic objects and “continuous” topological objects to gain information about the unstructured discrete objects in combinatorics. That said, the folks over at the nLab work on some crazy abstract stuff, which seems about as far away as possible from the day-to-day realities of graph theory or set systems. And maybe it is — but I hope it’s not, and as far as I’m concerned, this is a windmill that deserves to be tilted at. (After all, it might be a giant.)

So as my jumping-off point, I’ll take my observation from last time that the relationship between graphs and digraphs is analogous to the one between groupoids and categories. I briefly mentioned something called a quiver, which can be thought of as any of the following:

• Another name for a digraph, which categorical people use when they don’t want us combinatorialists stomping in and getting the floor all muddy;
• A “free category,” i.e., one in which there are no nontrivial relations between composition of morphisms;
• An algebraic object whose representations we want to consider; it’s worth thinking of this way mostly because of the “freeness,” although if you try to define it more formally you’ll probably end up with the previous definition;
• What you get when you take (part of) a category and forget all the rules for how morphisms compose.

This last point is the most interesting one for our purposes, since it’s clearly an algebraic object but isn’t as restrictive as “free category,” and thus has a chance of capturing the unstructured behavior of the combinatorial zoo. But it’s tricky to turn this into a rigorous definition that actually includes everything we want to be a quiver… so we’ll just use “quiver” as a fancy name for “digraph.” However, there’s an important philosophical lesson to be learned from the final point, so I’ll set it off:

Philosophical lesson. The edges of a quiver shouldn’t carry any information except for the vertices they are incident to; more generally, paths in a quiver shouldn’t carry any information except for their sequence of vertices.

### Generalized LYM inequalities

November 6, 2009

I’ve been thinking about this problem ever since an old post of Qiaochu’s first raised the question, and I’ve been frustrated by my inability to solve it. I could post it on MO, but I sort of already have, and anyway it raises questions which are too ill-formed right now to be right for MO. So anyway, here we go:

A lot of problems in extremal combinatorics correspond to finding large antichains in partially ordered sets. (By the way, all posets in this post will be assumed to have a least element.) Classically speaking, Dilworth’s theorem completely characterizes the size of antichains in posets; however, this is often tricky to apply, since it’s not always clear whether a partition into chains is minimal. In addition, it’s sometimes the case (particularly with infinite posets) that there are infinitely long antichains, but a nontrivial bound should still be attainable. The way to get by both of these obstacles is to assign weights to the elements of the poset, and rather than looking for large antichains, we look for antichains with high total weight.

The classical example of this solves the problem of finding the largest antichain in the lattice of subsets of a given finite set — the content of Sperner’s theorem. (more…)

### Category theory and combinatorics

June 30, 2009

This post is about two very different areas of mathematics that I’m pretty interested in (although I know a lot more about one of them…) and the relationships between them. It’s well-known that category theory grew out of investigations in algebra and topology, and that its insights are today still most useful in those and closely related subjects. (Algebraic geometry, I’m looking at you.) In particular, the famous “slum of topology” epithet notwithstanding, combinatorics is pretty far-removed from both the above areas, and indeed, I don’t know of an introductory textbook in combinatorics that gives category theory a second (or first!) glance.

And why should it? Doubtless many have tried, but no one’s (as far as I know) managed to give combinatorics a Grothendieck- or MacLane-type makeover. On one level, this makes sense; combinatorics is a huge area that encompasses problems that look completely different. It can almost be described as the trash-heap of mathematics; combinatorics is the study of problems that don’t really fit in anywhere else. To abstract from such a huge generality would seem sort of… onanistic, “abstraction for abstraction’s sake.” But of course this doesn’t mean that category theory has no place in combinatorics!

In this post, I plan to discuss two (and a half) different applications of category theory to combinatorics. (more…)