To avoid guessing which of arxiv preprints are likely to pass peer review and which are not, I will talk about theorems that were peer-reviewed and published in 2023. Most of them appeared in arxiv before.

I will list theorems that are at the same time great and have easy-to-understand formulation. So, you can just follow the links, read the original papers, and most likely you will be able to understand and appreciate these theorems! Enjoy!

So, the greatest easy-to-understand theorems published in 2023 are:

Proof that for every $\beta>\frac{1}{4}$, every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colourable, see https://doi.org/10.1016/j.aim.2023.109020 (this has then been improved to $O(t\log\log t)$, but this is not published yet, just in arxiv https://arxiv.org/abs/2108.01633). This is a progress towards famous Hadwiger's conjecture predicting that every graph with no $K_t$ minor is $(t-1)$-colourable.

Proof of the existence of polynomial automorphisms of ${\mathbb C}^2$ with wandering Fatou components, see https://doi.org/10.1090/jams/1005 This answers a question of Bedford and Smillie from 1991.

Proof that famous Schinzel's hypothesis H is true in 100% cases, in a well-defined sense, see https://doi.org/10.1007/s00222-022-01153-6

Proof of higher uniformity of bounded multiplicative functions in short intervals on average, see https://doi.org/10.1017/fmp.2023.28 and
https://doi.org/10.4007/annals.2023.197.2.3

Establishing the rigidity of proper colourings of ${\mathbb Z}^d$, see https://doi.org/10.1007/s00222-022-01164-3

Proof that if $n>d \geq 2$ and $(n,d)\neq (4,3)$, then $100\%$ of the $n$-variable forms of degree $d$ with rational coefficients (ordered by height) satisfy the Hasse principle. Also, if $(n,d)\neq (3,2)$, then a positive proportion of them has a non-trivial rational zero. See https://doi.org/10.4007/annals.2023.197.3.3

Proof that, in the black-box model of random polynomial systems with prescribed evaluation complexity $L$, we can compute an approximate zero of a random structured polynomial system with $n$ equations of degree at most $D$ in $n$ variables with only $poly(n,D)L$ operations with high probability. This exceeds the expectations implicit in Smale’s 17th problem, see https://doi.org/10.1017/fmp.2023.7

Proof that for any $d\geq 2$, any non-degenerate submanifold in ${\mathbb R}^d$ is of Khintchine type for convergence, see https://doi.org/10.4310/ACTA.2023.v231.n1.a1 This is a fundamental result in the theory of Diophantine approximation! See also https://doi.org/10.1007/s00222-022-01171-4 for another 2023 result in this area.

A new lower https://doi.org/10.1007/s00222-022-01177-y and upper https://doi.org/10.1007/s11425-022-2193-8 bounds for Erdos--Hooley delta-function that measures the maximal concentration of divisors of $n$ in short intervals

Proof of the Erdos primitive set conjecture https://doi.org/10.1017/fmp.2023.16

Proof that any multiple polylogarithm of weight $n\geq 2$ can be expressed as a linear combination of multiple polylogarithms of depth at most $n/2$ and products of polylogarithms of lower weight, see https://doi.org/10.1090/jams/1011

Proof of sharp isoperimetric inequalities for affine quermassintegrals, see https://doi.org/10.1090/jams/1013

A solution to Erdos and Hajnal's odd cycle problem https://doi.org/10.1090/jams/1018

Resolution of the Erdos-Sauer problem on regular subgraphs https://doi.org/10.1017/fmp.2023.19

Proof that random polynomials are irreducible with high probability https://doi.org/10.1007/s00222-023-01193-6

Resolution of Banach's isometric subspace problem in dimension four https://doi.org/10.1007/s00222-023-01197-2

Proof of the Erdos-Faber-Lovasz conjecture for all large $n$: there is a constant $n_0$ such that every linear hypergraph on $n\geq n_0$ vertices has chromatic index at most $n$, see https://doi.org/10.4007/annals.2023.198.2.2

Proof of the Erdos-McKay conjecture, which predicted that every $n$-vertex $C\log n$-Ramsey graph contains an induced subgraph with exactly $x$ edges for every not-too-large integer $x$, see https://doi.org/10.1017/fmp.2023.17

Verification of Polya's conjecture for Euclidean balls https://doi.org/10.1007/s00222-023-01198-1

Analysis of a multi-parameter variant of the Bellow-Furstenberg problem https://doi.org/10.1017/fmp.2023.21

Description of the equality cases in the Alexandrov-Fenchel inequality for convex polytopes https://dx.doi.org/10.4310/ACTA.2023.v231.n1.a3

Proof that any finitely generated field of characteristic $\text{char}(F) \neq 2$ can be characterized by a first-order sentence in the language of rings, see https://doi.org/10.4007/annals.2023.198.3.4

Proof that for every smooth Jordan curve $\Gamma$ and every cyclic quadrilateral $Q$ in the Euclidean plane, there exists a quadrilateral similar to $Q$ whose vertices lie on $\Gamma$, see https://doi.org/10.1007/s00222-023-01212-6

questiona useful source of significant results I wasn't aware of. It should be re-opened. $\endgroup$7more comments