55 problems

Cameron–Erdős conjecture.

Cameron's conjecture.

Lev's periodicity conjecture. If

Hamidoune–Plagne conjecture. The largest possible density is

Asymptotic sum-free-set conjecture. One should have

Half-density conjecture. For every symmetric, convex region $R\subset\mathbb{R}^d$,

Let $[n]=\{0,1,\ldots,n-1\}$, and let $c_d$ be the limiting density of a largest sum-free subset of $[n+1]^d$. For $u\in[0,d]$, define … and let $u_d^*$ maximize…

Let $w(n)$ be a function growing arbitrarily slowly, and let a sum-free subset of $\{1,\ldots,n\}$ be a set containing no two elements whose sum is also in the set. Structural conj…

Let a sum-free subset of $\{1,\ldots,n\}$ be a set containing no two elements whose sum is also in the set. Write $c_e$ and $c_o$ for constants, approximately $6.8$ and $6.0$, resp…

Elsholtz–Rackham's higher-dimensional conjecture. The number of sum-free subsets of $[n]^d$ is

Higher-dimensional extremal conjecture.

Elsholtz–Rackham's conjecture. The number of sum-free subsets of $[n]^2$ is

Let $G$ be a finite Abelian group of type I, and let $f^*_{\max}(G)$ denote the number of maximal distinct sum-free subsets of $G$. Define the growth of $f^*_{\max}$ relative to…

Let $G$ be an Abelian group of order $n$, and let $\mu(G)$ be the maximum size of a sum-free subset of $G$. Hassler–Treglown conjecture. Either … or … The paper constructs infinite…

Let $G$ be a finite Abelian group of type I, meaning that its order is divisible by a prime congruent to $2$ modulo $3$, and let $\mu(G)$ and $f_{\max}(G)$ denote the maximum size…

Let $G$ range over finite Abelian groups, and let $f_{\max}(G)$ denote the number of maximal sum-free subsets of $G$. Define the growth of $f_{\max}(G)$ as the smallest…

Let $G$ be a type $I$ abelian group of order $n$. Write $f^\star_{\max}(G)$ for the number of maximal distinct sum-free subsets of $G$, and let $\mu(G)$ denote the maximum density…

Let $k\bin\mathbb N$ and set $n:=5^k$. Write $f_{\max}(G)$ for the number of maximal sum-free subsets of a finite abelian group $G$. The elementary abelian 5-group conjecture. … Th…

Absolutely continuous coupling conjecture. For every $d\in\mathbb{N}$, there exist distributions $f_1,f_2,f_3,f_4$ on $\mathcal{C}$, absolutely continuous with respect to Lebesgue…

Compatible-uniform-coupling conjecture. For every $d\in\mathbb{N}$, the triple

The asymptotic density conjecture. For all $d\in\mathbb{N}$,

Calkin–Erdős conjecture. A $k$-sum-free subset of the non-negative integers has density at most

Classification conjecture. If

Let $G$ be a group, and let a maximal sum-free set be a sum-free subset of $G$ that is maximal under inclusion. Giudici–Hart conjecture. If $|G|>24$, then $G$ does not contain a ma…

Let $[1,n]=\{1,2,\ldots,n\}$. A subset of $[1,n]$ is partitionable into two sum-free sets if it can be written as the disjoint union of two sum-free subsets. Hancock–et al. conject…