3 problems

Let $G$ be a finitely generated group of exponential growth. Let $k\colon\mathbb{N}\to\mathbb{N}$ be nondecreasing with $k\ll\sqrt{|\mathbf{B}_S(n)|}$, and let $\mathsf{A}_k$ be a…

Let $k\ge 3$, $a>0$, and $D\ge 1$, and let $X$ count the edges in the random subhypergraph ${\mathcal H}_p$ of a $k$-uniform hypergraph under the hypotheses of the upper-tail theor…

Optimal sampling bound conjecture. A random subset $\Omega\subset\{0,1,\ldots,n-1\}$ of average cardinality