I am delighted to have found your blog. I would like to know whether you’d recommend some particular reading for someone interested in learning about graph limits (apart from the original articles by Lovász and collaborators). My first encounter with the formalism was through Chatterjee and Diaconis (Estimating and Understanding Exponential Random Graphs), and would like to read more about it. Thanks!

]]>Excellent post on regularity lemma and triangle removal lemma. ]]>

You need to know something about the pseudorandomness of the primes with respect to the patterns you’re trying to count, in order to apply a second method bound.

For example, Tao and Ziegler showed that dense subsets of the primes contain arbitrarily polynomial progressions, but we don’t know how to extend this result to higher dimensions, because there’s isn’t an extension of the Green-Tao-Ziegler result (on the asymptotic number of certain patterns in primes) for polynomials.

]]>It seems from these results that it’s basically possible to extend any conclusion that you want similar to the original Green-Tao theorem to higher dimensions, and to objects like positive density subsets of P^d.

Is this pretty universally true, or can you give an example of a statement of this nature that isn’t true? Where’s the subtle line?

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