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 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 .