We give new bounds for list sizes of random linear codes at capacity, max loads of linear hash functions, and Furstenberg sets, over every finite field $\mathbb{F}_q$.

1. Random linear codes over $\mathbb{F}_q$ with rate $1 - H_q(p) - \epsilon$ are $(p, O(q H_q(p)/\epsilon))$-list decodable with high probability for all ... more >>>

We prove that, for every constant $\varepsilon>0$ and every function $f:\Sigma^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the output equal to $b$ ... more >>>

We prove that for all $\varepsilon>0$, there exists a positive integer $k$ such that given a $4$-to-$1$ Game $\Psi$ with alphabet size at most $k$, it is NP-hard to distinguish between the case that val$(\Psi)=1$ and the case that val$(\Psi)\leq \delta$. This confirms the $4$-to-$1$ Games Conjecture from [Khot, CCC ... more >>>