26 problems

Chung–Goldwasser conjecture. Every measurable 3-sum-free subset $A$ of $[0,1]$ satisfies

Let $[N]=\{1,\dots,N\}$. A subset $A$ of the integers is sum-free if there do not exist $x,y,z\in A$ such that $x+y=z$. Cameron–Erdős conjecture. The number of sum-free subsets of…

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

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

Higher-dimensional extremal conjecture.

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 $G$ 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

Shakan's stronger conjecture. For every positive integer $S$, there exists $N_S\in\mathbb{Z}$ such that

Shakan's conjecture. For every positive integer $S$, there exists $N_S\in\mathbb{Z}$ such that

Asymptotic density conjecture. For positive integers $p,q$ and sufficiently large $n$,

Extremal bounds conjecture.

Let $k<\ell$ be positive integers. For a set $A$ of $N$ positive integers, let $\mathscr{M}_{(k,\ell)}(A)$ be the size of its largest $(k,\ell)$-sum-free subset, where a set is…

Let $A$ be a set of $N$ positive integers. A subset $\Omega\subseteq\mathbb{R}/\mathbb{Z}$ is maximal sum-free if it is sum-free and has measure $1/3$. Let $\mathbbm{1}_{\Omega}$ d…

Independence conjecture.

Exact-extremal-set conjecture. Let $n,r$ be positive integers with $r\geq4$. If $n$ is even and $r\leq7$, then

The restricted $(k,\ell)$-sum-free asymptotic conjecture.

Let $G$ be a finite abelian group of type I, and let $\mu(G)$ be the size of a largest sum-free subset of $G$, while $f_{\max}(G)$ is the number of maximal sum-free subsets of $G$.…

For an abelian group $G$, let $\mu(G)$ denote the size of a largest sum-free subset of $G$, and let $f_{\max}(G)$ denote the number of maximal sum-free subsets of $G$. Conjecture o…

Let $\mathcal{L}$ denote the equation $x+y=z$. A subset of $[n]$ is $(\mathcal{L},2)$-free if it admits a 2-colouring with no monochromatic solution to $\mathcal{L}$. Counting conj…

Let $\mu(G)$ denote the relevant group parameter used in the paper, and let $f_{\max}(G)$ denote the number of maximal sum-free subsets of a finite abelian group $G$. Few maximal s…

Let $G$ be a finite abelian group of order $n$, and let $f_{\max}(G)$ denote the number of maximal sum-free subsets of $G$. Universal bounds conjecture. For every such group, … whe…

Consider the random greedy independent set algorithm on a sufficiently nice hypergraph, meaning one that is almost uniform, almost regular, and not too sparse. Let $Q$ denote the n…