4 problems

Let $a_1G_1,\ldots,a_kG_k$ be pairwise disjoint left cosets in a group $G$, where each subgroup $G_i$ has finite index. Sun's finite-coset index conjecture. There are indices $i<j$…

Let $S(n,k)$ denote the number of partitions of an $n$-element set into exactly $k$ nonempty subsets, and let $v_2$ denote the $2$-adic valuation. For fixed $k\in\mathbb{N}$, consi…

Erdős–Stein conjecture.

Let $k,t\geq 2$ be coprime, and let $\prec_{k,t}$ be the ordering on $\mathbb{Z}/t\mathbb{Z}$ defined by the eventual comparison of the quantities $D_k^\times(r,t;n)$: namely,…