David Conlon pointed out to two remarkable papers that appeared on the arxive:
Joel Moreira solves an old problem in Ramsey’s theory.
Abstract: An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair 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 and .
Ernie Croot, Vsevolod Lev, Peter Pach gives an exponential improvement to Roth over .!
Abstract: We show that for integer , any subset of free of three-term arithmetic progressions has size , with an absolute constant 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 result quite mind boggling! What does it say about ???
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.