David Conlon pointed out to two remarkable papers that appeared on the arxive:

### Joel Moreira solves an old problem in Ramsey’s theory.

Monochromatic sums and products in $\mathbb N$.

Abstract: An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair ${x+y,xy}.$ We answer this question affirmatively in a strong sense by exhibiting a large new class of non-linear patterns which can be found in a single cell of any finite partition of N. Our proof involves a correspondence principle which transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as $x^2-y^2=z$ and $x^2+2y^2-3z^2=w$.

### Ernie Croot, Vsevolod Lev, Peter Pach gives an exponential improvement to Roth over $\mathbb Z_4^n$.!

Abstract: We show that for integer $n>0$, any subset $A$ of $Z^n_4$ free of three-term arithmetic progressions has size $|A|<2(\sqrt n +1)4^{cn}$, with an absolute constant $c$ approximately equal to 0.926.

David Ellis made a few comments: Three ‘breakthrough’ papers in one week – one in combinatorial geometry (referring to the Erdos-Szekeres breakthrough) , one in additive number theory and one in Ramsey theory – not bad!; . I’ve now read all of the proofs and am sure (beyond reasonable doubt) that they’re all correct – an unusually short time-frame, for me at any rate! The Croot-Lev-Pal Pach paper is a really beautiful application of the polynomial method – a ‘genuinely’ self-contained paper, too, and very nicely written.

I find the $Z_4^n$ result quite mind boggling! What does it say about $Z_3^n$???

### Post by Green

In another facebook post Ben Green writes:  I Wish I could just casually hand Paul Erdos a copy of Annals of Math 181-1. 4 of the 7 papers are: solution to the Erdos distance conjecture by Guth and Nets Katz, solution to the Erdos covering congruence conjecture by Hough, Maynard’s paper on bounded gaps between primes, and the Bhargava-Shankar paper proving that the average rank of elliptic curves is bounded.

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### 4 Responses to More Math from Facebook

1. gowers says:

Well, now we know what it says about $\mathbb Z_3^n$

• Gil Kalai says:

Amazing. But then, what does it say for $r_3(n)$ 🙂 ?