We prove that any subset $A \subseteq [3]^n$ with $3^{-n}|A| \ge (\log\log\log\log n)^{-c}$ contains a combinatorial line of length $3$, i.e., $x, y, z \in A$, not all equal, with $x_i=y_i=z_i$ or $(x_i,y_i,z_i)=(0,1,2)$ for all $i = 1, 2, \dots, n$. This improves on the previous best bound of $3^{-n}|A| ... more >>>

Let $\Sigma_1,\ldots,\Sigma_k$ be finite alphabets, and let $\mu$ be a distribution over $\Sigma_1 \times \dots \times \Sigma_k$ in which the probability of each atom is at least $\alpha$. We prove that if $\mu$ does not admit Abelian embeddings, and $f_i: \Sigma_i \to \mathbb{C}$ are $1$-bounded functions (for $i=1,\ldots,k$) such that ... more >>>

We prove local and global inverse theorems for general $3$-wise correlations over pairwise-connected distributions. Let $\mu$ be a distribution over $\Sigma \times \Gamma \times \Phi$ such that the supports of $\mu_{xy}$, $\mu_{xz}$, and $\mu_{yz}$ are all connected, and let $f: \Sigma^n \to \mathbb{C}$, $g: \Gamma^n \to \mathbb{C}$, $h: \Phi^n \to ... more >>>