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Appearances are deceiving. DHJ is a deep theorem. It turns out to have as a consequence many other important and hard-to-prove results of mathematics, some in areas that appear quite unrelated. Think of it as a domino: when it falls, it causes many other important and otherwise hard-to-budge mathematical dominoes to also fall. Let me give you an example of the way DHJ connects to another part of mathematics that seems unrelated—the problem of understanding the structure of the prime numbers. It turns out that DHJ implies a deep result of number theory called Szemerédi’s theorem. That theorem was first proved in 1975 by the mathematician Endre Szemerédi; mathematicians have since found several additional proofs. Using ideas drawn from several of those proofs, in 2004 the mathematicians Ben Green and Terence Tao proved a major new result about the structure of the prime numbers. To understand what the Green-Tao theorem says, consider the sequence of numbers 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089. These are all prime numbers, and they’re evenly spaced; each member of the sequence is 210 larger than the one that precedes it. What the Green-Tao theorem says is that you can find evenly spaced sequences of prime numbers of any length whatsoever. Want an evenly spaced sequence of a million prime numbers? Green-Tao guarantees that such a sequence exists. The theorem doesn’t actually give an easily usable recipe for finding such a sequence, but it guarantees that if you search for a sequence long enough, you’ll find it eventually. Now, results about the prime numbers probably seem quite unrelated to worrying about line-free configurations in high dimension. And yet the DHJ-Szemerédi and Szemerédi–Green-Tao connections suggest that there really is a connection between DHJ and the structure of the prime numbers.

The DHJ theorem was first proved in 1991 by the mathematicians Hillel Furstenberg and Yitzhak Katznelson. So when Tim Gowers proposed the Polymath Project, he wasn’t proposing that the polymaths find the first proof of DHJ. Rather, he was proposing that they find a new proof. You may be surprised that a top mathematician such as Gowers would be interested in finding a new proof of an already known result. But the existing proof of DHJ used indirect and rather advanced techniques from a branch of mathematics called ergodic theory. While it was a perfectly good proof, Gowers believed that additional insight into DHJ could be gained by finding a new proof that relied on different techniques. In particular, Gowers was interested in finding a proof that relied only on elementary techniques, that is, techniques that didn’t require sophisticated mathematics such as the tools of ergodic theory. Sometimes, finding new proofs can give us significant new insights that help us understand why a result is true in the first place. Indeed, this is exactly what happened with the multiple proofs of Szemerédi’s theorem. When Green and Tao proved their theorem about prime numbers, they drew on ideas from several different proofs of Szemerédi’s theorem. That made finding a new proof of the DHJ theorem using only elementary techniques a challenging and worthwhile goal for the Polymath Project.

Acknowledgments

In writing this book I’ve be

nefited enormously from the enthusiasm, insight, and support of many people. Especial thanks to Peter Tallack, my agent, whose enthusiasm for the project, perceptive feedback, and knack for asking the right questions has dramatically improved the book. Many thanks also to the team at Princeton University Press, both for their enthusiasm, and for patiently helping me turn this book into a reality. I’m particularly grateful to my editor, Ingrid Gnerlich, as well as Jodi Beder, Bob Bettendorf, Christopher Chung, Kathleen Cioffi, Peter Dougherty, Jessica Pellien, and Julie Shawvan. Thanks to Simon Capelin, Kelly McNees, and Lee Smolin for helpful comments on early versions of