# Blog

## Papers by MIT combinatorialists—Fall 2023

December 22, 2023

As we near the end of the year, I’m delighted to showcase and celebrate a selection of recent papers by the talented combinatorialists at MIT, especially those by students and postdocs. These papers reflect the culmination of their hard work, dedication, and innovative problem solving. For each paper, I’ll offer a concise summary, and I encourage you to read the papers to learn more about the beautiful mathematics within.

This paper proves the first quantitative upper bound on an interesting special case of the polynomial Szemerédi theorem. Specifically, it shows that the maximum size of a subset of $\{1, \dots, N\}$ avoiding the pattern $x, x + y^2 - 1, x + 2(y^2 - 1)$ is at most $N / \log \log \cdots \log N$, with at most a constant number of logs. This special case was highlighted by Green.

This paper introduces a new perspective on the polynomial method, unifying various earlier arguments in the field that initially looked quite distinct.

This paper determines the density of the largest sum-free subset of the lattice cube $\{1, \dots, n\}^d$ for $d = 3, 4, 5$, resolving a conjecture of Cameron and Aydinian in these dimensions.

This paper proves that over the $p$-adics and profinite integers, any set which contains a 2-flat in every direction must have positive measure. This extends earlier results on discrete Kakeya.

This paper shows that the maximum size of a subset of $\{1, \dots, N\}$ without a 5-term arithmetic progression is at most $N \exp ( -(\log\log N)^c)$, improving Gowers’ longstanding $N (\log\log N)^{-c}$ bound.

### Discrete Geometry

This paper shows that every subset of a sphere avoiding an orthonormal basis has measure at most $c_0$, where $c_0 < 1$ is some absolute constant independent of the dimension.

Extending earlier work on equiangular lines, this paper studies the maximum number of unit vectors in $\mathbb{R}^d$ whose pairwise inner products lie in some given $L \subseteq [-1,1)$, specifically in the case where $L = [-1, -\beta] \cup \{\alpha\}$. The main result determines all such $L$ where the “uniform bound” of Balla, Dräxler, Keevash, and Sudakov is tight.

### Random Matrix Theory

Let $A_n$ be an $n \times n$ random matrix whose entries are independent Bernoulli random variables with probability $d/n$ (where $d > 0$ is a constant). This paper shows that the spectral distribution $\mu_{A_n}$ converges in probability to some limiting distribution $\mu_d$ on the complex plane.

### Random Graphs

This paper studies the structure of a sparse Erdős–Rényi random graph conditioned on a lower tail subgraph count event. When the subgraph count is conditioned to be slightly below expectation, the typical random graph is shown to be close to another Erdős–Rényi random graph (with lower edge density) in both cut norm and subgraph counts.