Most of what I could say has already been said better, so I’ll link to some other tributes shortly. People have talked about how Big Star were a major influence on all sorts of bands in the ’80s and ’90s up to the present, from R.E.M. to the Replacements to Elliott Smith. People have talked about the timeless nature of Big Star’s songs, about how “Thirteen” is maybe the ultimate expression of what it’s like to be that age. But one thing that’s perhaps been lost in the chatter is the fact that so much of Chilton’s work with Big Star is just plain wonderful music. Witness “Nighttime” and “Blue Moon,” back-to-back tracks from Big Star’s final album, Third (also known as Sister Lover) — to my ears, two of the most achingly beautful pop songs ever recorded.

The best tribute I’ve read — maybe the definitive one — is Paul Westerberg’s in the New York Times. (Westerberg wrote and performed, with the Replacements, the song that gave this post its title.) But the best possible tribute, in my mind, would be if this blog post turned one person on to the music of the late, great Alex Chilton. It might change your life. More likely it won’t. But either way, it’s great damned music, and I hope — and expect — that it’ll live on far after I’m gone.

Theorem. Graphs of order and girth at least 5 have edges.

Proof. Suppose that has edges. Define the function to be the “adjacency characteristic function.” Now, by Cauchy-Schwarz:

.

Clearly for fixed, is unbounded as . But the first expression is times the number of (possibly degenerate) 4-cycles in the graph; it’s easy to check that there are degenerate 4-cycles, so as n is unbounded our graph must contain a 4-cycle. QED

Now, it’s possible to get better bounds by cleverly doing “surgery” on the graph and just using pigeonhole. (See here for details.) But it’s tricky and far less beautiful than the argument with Cauchy-Schwarz, which really demonstrates one way in which C-S can be thought of as “strengthening” pigeonhole.

Proof. Consider a tree on n vertices with root . Assign to each edge an element of the vector space , 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 sum to 0. There is a natural “distance” function on E w/r/t ; let have maximal distance in S, and suppose WLOG that t is farther than s from . 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 of them.

Okay, so this seems like a lot of work to establish something that can be done in two lines with induction. And in fact you can view the above proof as inductive as well; if we build edge by edge, we get the necessary statement that |E| = n-1 as a bonus.

But of course that can also be established by topological methods, which means that the only “combinatorial” part of the whole proof lies in picking the basis of our vector space. And in fact that’s really all we need, the rest of the proof being more or less window dressing:

Proof (of Theorem). Let . Pick a basis of V. Proceed as in the end of the first proof.

So the first proof is pretty much useless. However, there are at least two ways it can be modified to be made (potentially!) interesting.

First, note that the construction of the vector space and independent set in the first proof was, in fact, a construction of a representation for the cycle matroid of T over . It’s a famous theorem that every graphic matroid is representable over $latex  GF(2)$ (and, in fact, over every field) — if this theorem, or more likely a purely matroid-theoretic version, can be proved without recourse to induction or linear algebra, we might obtain a “new” proof that .

Second, the cycle matroid of a simple graph is a simple matroid. In particular, if we consider the representation over of the cycle matroid of analogous to the one we constructed, we have an immediate proof that .

Questions: Can we give a “purely matroid-theoretic” proof of ? Can we give a matroid-theoretic proof that exponential functions eventually dominate polynomial functions?

(Hat tip to Vladimir S. for the bad pun; new post coming soon, really, I promise.)

Smithers: I’ve got to find a replacement who won’t outshine me. Perhaps if I search the employee evaluations for the word ‘incompetent…’

[Computer beeps, screen displays "714 Matches Found']

Smithers: 714 names?! Heh. Better be more specific. ‘Lazy,’ ‘clumsy,’ ‘dimwitted,’ ‘monstrously ugly.’

[After a couple seconds, computer beeps, screen displays '714 Matches Found']

Smithers: Ah, nuts to this, I’ll just go get Homer Simpson.

Actually, I tried to find the statistical nomenclature for this kind of thing, but couldn’t. Anyone have any idea what this is? (I want to say selection bias, but that’s not quite it…)

New Extremal Toolbox post should be coming later this weekend.

Full sequence in a squarish table:

Now the HAPs as columns:

first column is the sequence, second column is x_{2n},…

Hence, “The Extremal Toolbox.” In each post, I’ll take a (solved!) problem in extremal combinatorics — anything from Sperner’s theorem to Kakeya over finite fields, as long as there’s an extremal flavor — and try to break down a proof into its component parts.

Today I’m going to examine a problem which appeared on MathOverflow some time ago, which I didn’t quite solve (but came within epsilon of!) The relevant post is here; if you don’t care to click through, here’s the problem.

Let be an matrix with non-negative integer entries. Suppose further that if is 0, then the sum of all the entries in the ith row or the jth column is at least . Then the sum of all the entries in is at least .

How do we solve this?

The first thing to notice is that (as long as n is even) there are a number of matrices with total sum exactly — for instance, a “checkerboard pattern” of zeroes and ones, or having all entries along the diagonal be equal to n/2 and zeros everywhere else. There’s no apparent structure to these extremal entries… but one thing that you do notice is that all of them have row sums and column sums at least . Note that if we can prove that the entries in any row and column have to sum to at least , we’re immediately done, since there are n rows.

Trick: Write the quantity you want to optimize as a sum of “sub-quantities” which are easier to analyze.

New goal: Show that each row/column has sum at least n/2.

The obvious path here is to try for a proof by contradiction. Suppose that some row, say, has sum . Then since our matrix has integer entries, that row has a lot of entries which are zeros; the columns containing those zeros each have to have entries summing to at least . This looks like it could work!

Let’s make it more rigorous. Let’s say that the row has total sum , for some . Then the row must contain at least zeros; the columns containing these zeros have to each have sum at least , so the matrix must have sum at least . But as c approaches 1/2, this lower bound only gives us , which is obtainable by a number of methods.

But… we haven’t used all our information! In particular, we know that there’s a row whose entries sum to . And since the choice of row was arbitrary, we can even assume that it was the row with the smallest sum! Does this let us do better?

It sure does — following the above argument line-by-line, we have that the sum of all the entries is the maximum of — doing some high-school algebra, we see that the coefficient of this maximum will always be at least , which is about 0.38. This is a major improvement on 0.25, but it’s still not what we want.

Trick: If you have a free parameter in your argument, try making it extremal. The worst case scenario is that you gain no additional information; the best case is a considerably stronger result.

Darn. Is there any way to rescue this approach?

Well, look back over the argument. Is there anywhere where there’s some slack, where we can tighten it up? At first it doesn’t look like it — we’ve optimized and extremized everything we could. But note that so far we’ve looked at row sums and only row sums. Of course, it doesn’t really matter — the argument would proceed exactly the same if we used column sums — but “Without Loss of Generality” is a tricky phrase, especially in combinatorics, where it often means you’re giving up some tightness for ease of analysis.

So let’s try, instead of minimizing over all row sums, minimizing over all row sums and column sums. Suppose that there’s a row (WLOG, again, but this time we’re getting more information) with total sum , such that every row and column of our matrix has total sum at least m. Now, again, there must be at least zeros in the row, and the sum of the entries in all the columns containing those zeros is at least . But this time we have some additional ammo; we know that the sum of the entries in the other columns is at least . So the sum of all the entries in the matrix must be at least ; it’s easy to check that the coefficient must always be at least 1/2.

Trick: If you have a free parameter, make it extremal — and keep in mind that it’s not always immediately clear how to do so! In general, be very careful when you’re trying to find the “slack” in your argument; it can be hidden.

What are the general lessons?

In actuality, this problem wasn’t very hard — the only real trick is considering the sums of the entries of the individual rows/columsn, which is an obvious enough thing to do. The key, though, was to make the argument as tight as possible, and not lose any information along the way. Sometimes this isn’t feasible; sometimes the best you can do is obtain an asymptotic bound. But when it is feasible — in particular, when it’s easy to get the right asymptotics without much thought at all — it’s often possible to work wonders with a fine-tuned version of the “obvious” approach.

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?

Unfortunately the question seems impossible to answer precisely — even today in some countries there’s no reliable system to record births and demographic data, and 100 years ago the situation was far worse. But we can certainly try to make an educated guess, or at least think about how we’d go about trying to make an educated guess!

So we’ll start off with our simplifying assumptions. First of all, in the analysis we’re going to have to consider the probability that a person who’s lived to N days will live to N+1 days. (Or a coarser version of this statistic.) Obviously this probability is different for each person — a 104-year-old in Bangladesh with a terminal disease and without access to good medical care has a much shorter life expectancy than a healthy person of the same age in, say, New Jersey. But this complicates matters hugely, so we’ll say that for each person the probability is the same.

In addition, we’ll assume that the birth rate was constant, say, 1900 and 1910, and that someone born in 1902 had the same probability of living to 100 and someone born in 1909. Again, these are simplifying assumptions.

So once you’ve made these assumptions, you end up with a sequence of random variables that describe how many people who have lived exactly N days are still living. Under the above (unrealistic) assumptions, is approximately a binomial distribution, with the probability decreasing exponentially with N.

So the first estimate we can make is to see when the expected value . But this isn’t all that good — probably the first N with came way earlier — and so a better thing to do is to estimate

.

But, when the are binomial, this probability is easy — it’s just for some fixed probability p and integer M. So the sum is

.

As long as this is small, the probability  is high that none of the are 0 for $j \leq N$.

Can we do better than this? What if we replaced the above model by something more realistic, where the death rate increases as N increases?

What’s (approximately) the most recent day no one alive was born?

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 of the question — what does it even mean, “local data?”

Gjergji left the meaning of “local graph property” vague in his question — and to his great credit, he stated (in the question!) that he did so knowingly and intentionally. To see why, let’s go over some theorems (and some conjectures) that can reasonably be said to “go from local to global” (for conciseness, we’ll set and in what follows):

1. If every two vertices in a connected graph G have a unique common neighbor, then there’s a vertex adjacent to all the others.
2. (Ore) If every pair of non-adjacent vertices in G has total degree at least n, then G is Hamiltonian.
3. If the subgraph spanned by every k vertices is bipartite, then G has chromatic number at most for some absolute constant c.
4. (Grotzsch) If G is planar and has no triangles, then G is 3-colorable.
5. (Dvorak-Kral-Thomas) If G is planar and every path between distinct triangles is longer than some absolute constant, then G is 3-colorable.
6. If no two vertices of G, both of degree greater than 2, are adjacent, then G is 3-colorable.
7. If G has roughly edges and about the same number of subgraphs isomorphic to as the Erdos-Renyi graph , then all labeled graphs on k vertices occur about equally often as induced subgraphs of G, for .
8. (Turan) If G is triangle-free, then .
9. (Robertson-Seymour) The graph minors theorem.
10. If G contains no subgraph and , then the complement of G must contain a $K_k$ subgraph.
11. (Alon-Krivelevich) If a random induced subgraph of G of order is k-colorable, then with high probability you can delete edges of G and obtain a k-colorable graph. (In particular, if every such subgraph is k-colorable, then the conclusion holds deterministically.)
12. (Conjecture, Erdos-Gyarfas) If G has minimum degree 3, then G contains a cycle with length a power of two.
13. (Conjecture, Hadwiger) If G contains no minor, then G is (k-1)-colorable.
14. (Reconstruction conjecture) As long as n > 2, G is determined by the multiset of induced subgraphs on (n-1) vertices.

Clearly these statements are all over the place — some, like 6, are almost trivial to prove, while others take hundreds of pages or are still open! Furthermore, they run the gamut from extremal graph theory to topological problems to pure combinatorics.

And implicitly they use widely varying notions of “locality.” Let’s try to deconstruct what they mean, in roughly increasing order of generality. But first, let’s clear up what we mean by “local-to-global,” at least a little.

The starting idea — although we may well move beyond this later — is to consider two graph properties . A local-to-global theorem, then, is a theorem which says that if G satisfies property P “locally,” then it satisfies P’ globally. (We may also require that G is sufficiently large, or dense.) Often P’ will be weaker than P, although sometimes they’re logically incomparable. The question is what we mean by “locally!” So let’s consider some possibilities:

• The “local data” are the induced subgraphs on a constant (or << n) number of vertices. This is arguably the strictest possible notion of “locality” we’ll encounter, and is almost certainly the strictest one about which we can say much of interest. Amazingly, though, there are a lot of interesting things to consider! Ramsey theory falls squarely into this category, as does a good deal of extremal graph theory. Statements 3, 8, 10, and the deterministic version of 11 are of this form. 7 and the probabilistic version of 11 are closely related.
• The “local data” are (unions of) the neighborhood of each vertex, the set of vertices at most a fixed distance away from each vertex, or the induced subgraphs on these. This isn’t actually comparable to the first notion of “locality,” but in practice it tends to be less strict. What’s interesting is that there seems to be a “phase transition” at work here: Theorems that just use the neighborhood of sets of vertices tend to be straightforwardly combinatorial; examples include 1, 2, 6, and 12. But theorems that consider higher distances often have a geometric flavor; look at (for instance) 5, or at the spectral theory of expanders.
• The “local data” are “low-complexity (induced) subgraphs.” Occasionally one wants to consider a family of graphs that isn’t closed under taking subgraphs, but are still in some sense “small.” One example, for instance, is the set of subdivisions of ; another is the set of odd cycles. It might be useful to think of this as an “information-theoretic relaxation” of small induced subgraphs; we can view the family of constant-sized subgraphs as the set of subgraphs describable with O(1) bits; this relaxes it to some family of subgraphs with smal information complexity. Probably the best example is graph minor theory, although the strong perfect graph theorem falls into this category as well.

Now let’s step back from the original definition of “local-to-global” and generalize. Note that if we can check a global property by checking it on each constant-sized subgraph, we can check it using a constant number of pointers and constant extra space. This suggests that we might want to investigate “complexity-theoretic” notions of localness.

• A property is “locally determined” if it’s decidable by a Turing machine with a constant number of pointers to the input and constant additional space. I believe this is equivalent to the property being regular, although I’m a little unclear.
• A property is “locally determined” if it’s in L = SL. Think of this as a generalization of “counting on local data.” This is a remarkably rich complexity class, containing things like planarity and connectedness that we might not think of as “local.”
• A property is “locally determined” if it’s in RL. Similar to the above.
• A property is “locally determined” if it’s in P (=BPP? =NP?). The rationale for this is that no such polynomial time algorithm can examine more than a negligible fraction of “large subgraphs.” That said, I don’t know of any graph property that I would consider “local” which is known to be in BPP but believed to be outside of L. Anyone have an example?
• ??? What are some other notions of “local” in graph theory? How does this affect how we move from local to global?