12 problems

Let $p$ be a prime, and consider pairs of positive integers $(m,n)$ in a reasonable planar region contained in the first quadrant. The relevant density is the probability that…

Let $p_i$ denote the $i$th prime. Let $C_k(n)$ be the subsets of $\{1,\ldots,n\}$ containing no $k+1$ pairwise coprime integers, and let $E_k(n)$ be the integers at most $n$ divisi…

Let $p_i$ denote the $i$th prime and set $P_k=p_1p_2\cdots p_k$. Let $C_k$ be the family of subsets of the positive integers containing no $k+1$ pairwise coprime integers, and let…

Let $\mathbb{Z}^{+}$ be the positive integers, let $p_i$ be the $i$th prime, and let $C_k$ be the family of subsets of $\mathbb{Z}^{+}$ containing no $k+1$ pairwise coprime integer…

Let $\mathbb{Z}^{+}$ be the set of positive integers, and let $p_i$ denote the $i$th prime. Let $C_k$ be the family of subsets of $\mathbb{Z}^{+}$ containing no $k+1$ pairwise copr…

Endpoint equality conjecture. Then

Coprime-endpoint bound conjecture. For every such $k,l,a$,

Endpoint uniqueness conjecture. If

Eventual uniqueness conjecture. For every integer $k\ge3$, there exists an integer $n_k$ such that, for all $n\ge n_k$, if

Shifted-block bound conjecture. For every such $k,l,a$,

Primorial-threshold conjecture. For every $k\ge3$, setting

Erdős's conjecture. For all $n\ge p_k$, one has