Color each positive integer with red and blue, whether there must be three numbers $a, b, c$ with same color such that $a^2+b^2 = c^2$? [duplicate]

As the year is slowly coming to an end, I was wondering which great advances have there been in mathematics in the past 12 months. As researchers usually work in only a limited number of fields in mathematics, one often does not hear a lot of news about advances in other branches of mathematics. A person who works in complex analysis might not be aware of some astounding advances made in probability theory, for example. Since I am curious about other fields as well, even though I do not spend a lot of time reading about them, I wanted to hear about some major findings in distinct fields of mathematics.

I know that the question posed by me does not allow a unique answer since it is asked in broad way. However, there are probably many interesting advances in all sorts of branches of mathematics that have been made this year, which I might have missed on and I would like to hear about them. Furthermore, I think it is sensible to get a nice overview about what has been achieved this year without digging through thousands of different journal articles.

Personally, I was kind of fascinated by the solution to the Boolean Pythagorean triples problem which was finally solved in May. The problem asked whether or not the set of natural numbers $\mathbb{N}$ can "be divided into two parts, such that no part contains a triple $(a, b, c)$ with $a^2+b^2=c^2$". Heule, Kullmann and Marek managed to prove (with the help of a lot of computing power) that this is in fact not possible.

References:

Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016-05-03). "Solving and Verifying the Boolean Pythagorean Triples problem via Cube-and-Conquer".

Don't know if this counts, as the proof was announced in late 2015. Tao's solution of the Erdős discrepancy problem was published in 2016. You can find it here; it was actually the first paper of the Discrete Analysis journal.

For me, I was stunned at the discovery that: Prime numbers have decided preferences about the final digits of the primes that immediately follow them. It was found that among the first billion primes, for instance, a prime ending in $9$ is almost $65$ percent more likely to be followed by a prime ending in $1$ than another prime ending in $9$. This groundbreaking research has been done by Soundararajan and Lemke Oliver.

They believe that they have found that the biases they uncovered in consecutive primes come very close to what the prime k-tuples conjecture predicts. In other words, the most sophisticated conjecture mathematicians have about randomness in primes forces the primes to display strong biases.

You can read their paper here.

I don't really follow major breakthroughs, but my favorite paper this year was Raphael Zentner's Integer homology 3-spheres admit irreducible representations in $SL_2(\Bbb C)$.

It has been known for quite some time, and is a corollary of the geometrization theorem, that much of the geometry and topology of 3-manifolds is hidden inside their fundamental group. In fact, as long as a (closed oriented) 3-manifold cannot be written as a connected sum of two other 3-manifolds, and is not a lens space $L(p,q)$, the fundamental group determines the entire 3-manifold entirely. (The first condition is not very serious - there is a canonical and computable decomposition of any 3-manifold into a connected sum of components that all cannot be reduced by connected sum any further.) A very special case of this is the Poincare conjecture, which says that a simply connected 3-manifold is homeomorphic to $S^3$.

It became natural to ask how much you could recover from, instead of the fundamental group, its representation varieties $\text{Hom}(\pi_1(M), G)/\sim$, where $\sim$ identifies conjugate representations. This was particularly studied for $G = SU(2)$. Here is a still-open conjecture in this area, a sort of strengthening of the Poincare conjecture: if $M$ is not $S^3$, there is a nontrivial homomorphism $\pi_1(M) \to SU(2)$. (This is obvious when $H_1(M)$ is nonzero.)

Zentner was able to resolve a weaker problem in the positive: every closed oriented 3-anifold $M$ other than $S^3$ has a nontrivial homomorphism $\pi_1(M) \to SL_2(\Bbb C)$. $SU(2)$ is a subgroup of $SL_2(\Bbb C)$, so this is not as strong. He does this in three steps.

1) Every hyperbolic manifold supports a nontrivial map $\pi_1(M) \to SL_2(\Bbb C)$; this is provided by the hyperbolic structure. 2) (This is the main part of the argument.) If $M$ is the "splice" of two nontrivial knots in $S^3$ (delete small neighborhoods of the two knots and glue the boundary tori together in an appropriate way), then there's a nontrivial homomorphism $\pi_1(M) \to SU(2)$. 3) Every 3-manifold with the homology of $S^3$ has a map of degree 1 to either a hyperbolic manifold, a Seifert manifold (which have long been known to have homomorphisms to $SL_2(\Bbb C)$, or the splice of two knots, and degree 1 maps are surjective on fundamental groups.

The approach to (2) is to write down the representation varieties of each knot complement and understand that the representation variety of the splice corresponds to a sort of intersection of these representation varieties. So he tries to prove that they absolutely must intersect. And now things get cool: there's a relationship between these representation varieties and solutions to a certain PDE on 4-manifolds called the "ASD equation". Zentner proves that if these things don't intersect, you can find a certain perturbation of this equation that has no solutions. But Kronheimer and Mrowka had previously proved that in the context that arises, that equation must have solutions, and so you derive your contradiction.

This lies inside the field of gauge theory, where one tries to understand manifolds by understanding certain PDEs on them. There's another gauge-theoretical invariant called instanton homology, which is the homology of a chain complex where the generators (sorta) correspond to representations $\pi_1(M) \to SU(2)$. (The differential counts solutions to a certain PDE, like before.) So there's another question, a strengthening of the one Zentner made partial progress towards: "If $M \neq S^3$, is $I_*(M)$ nonzero?" Who knows.