Posts Tagged ‘MathOverflow’

The extremal toolbox: A matrix problem

January 10, 2010

I’m starting a new series of posts this semester where I get “back to basics.” One of the few areas of mathematics in which I can claim anything even in the same connected component as “expertise” is extremal combinatorics. Unfortunately for me and my lazy, big-picture brain, though, extremal combinatorics is very much a “problem-solving” subject, with a relatively small number of tools that are used to solve all sorts of different problems. So without some practice solving these problems, or expositing the solutions, it’s easy to get rusty.

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 M be an n \times n matrix with non-negative integer entries. Suppose further that if m_{ij} is 0, then the sum of all the entries in the ith row or the jth column is at least n. Then the sum of all the entries in M is at least n^2/2.

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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?”

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