115 problems

Let $j$ and $k$ be nonnegative integers with $k$ even. Prescribed-defect existence conjecture. There exists a positive integer $n$ and a set $S\subset\{0,1,\dots,n-1\}$ such that…

For $n\ge 1$, let $S$ range over subsets of $\{0,1,\dots,n-1\}$. For nonnegative integers $j$ and $k$, define … assuming the limit exists. Since $S-S$ is symmetric about $0$, only…

Limiting-proportion conjecture. The three limiting proportions all exist and are positive.

For a finite set $E$ of squares, write $E+E=\{x+y:x,y\in E\}$. Ruzsa's conjecture. For every $\epsilon>0$, … The source states that Chang's conjecture implies this one, while the B…

Structural conjecture. There exists a $K\in\mathbb N$ such that for every finite set $A\subseteq\mathbb N$ with $|A|>K$, if

Freiman's conjecture. There exists a natural number $K$ such that for any finite set of natural numbers $A$ with $|A|=k>K$ and

The sumset obstruction conjecture. If, for some $\kappa$, one has

Let $\mathscr P$ denote the set of all primes. For sets $S,T$ of positive integers, define $S\sim T$ to mean that their symmetric difference is finite. Ostmann's inverse Goldbach c…

Sharp sumset lower-bound conjecture. The inequality

For $cdelta cin (0,1)$ and $Ncinmathbb{N}$, let $cphi_{cdelta}(N)$ be the largest integer such that every set $Acsubseteq [N]$ with $|A|cgeq cdelta N$ contains a sumset $B+C$, wher…

Let $d\geq 2$ be fixed, let $q$ be a prime power with $q\equiv 1\pmod d$, and let $G\subseteq {\mathbb F}_q^*$ be the multiplicative subgroup of index $d$. Generalized Sárközy conj…

Let $A,B \subseteq ewline \mathbb Z$ be nonempty, and let $\mathbf F=(Fn){ninmathbb N}$ be a Følner sequence of finite subsets of $\mathbb Z$, meaning that … for every $t\in\mathbb Z$.…

Let $t$ be a positive integer, let $G$ be an abelian group, and let $A,B\subseteq G$ be finite subsets with $|A|,|B|\geq t$. Write $A+_iB$ for the set of elements of $G$ having at…

Generalized sumset pattern conjecture. There exists an infinite set $B\subset\mathbb N$ such that, for every $i=1,\ldots,k$,

$3k-4$ conjecture. If

Let $A\subset\mathbb{F}_{2}^{n}$ be a set with $|A+A|\leqslant K|A|$. Polynomial Freiman-Ruzsa conjecture in characteristic two. Then $A$ is covered by at most $2K^{C}$ cosets of s…

Let $A\subseteq\mathbb{Z}$ be a finite set, let $|A|=k$, and write $\operatorname{diam}(A)=\max A-\min A=d$. For $hA=\{a_1+\cdots+a_h:a_i\in A\}$, extremal diameter conjecture. If…

Let $\mathcal{R}(h,k)$ denote the set of possible cardinalities $|hA|$ for $k$-element sets $A$ of integers, and let $\Delta_{h,k}$ be the exceptional set defined in the paper. Pos…

Let $p$ be a prime, let $k$ be a positive integer, and let $\lambda_1,\dots,\lambda_k$ be nonzero integers viewed in $\mathbb{F}_p$. For a set $A\subseteq\mathbb{F}_p$, write…

Let $A$ be a set of size $s$ with doubling constant bounded by $K$, meaning $|A+A|\leq K|A|$. For each $ell$ let $\mathcal{F}_\ell$ be a collection of sets, and let $\tilde{O}$ and…

Krachun–Petrov conjecture. For every finite subset $A\subset\mathbb{C}$,

Concentration conjecture. There is an integer $n_0$ such that, for every $n\ge n_0$, at least $(1-\epsilon)\binom{n}{4}$ of the four-element subsets of $[n]$ have $h_1>h$; among th…

Mohan–Pandey conjecture. The following implications hold:

Freiman–Lev conjecture. One has

Let $A$ be a set of $k\geq 4$ positive integers, and let $h$ be an integer with $3\leq h\leq k-1$. The restricted signed $h$-fold sumset is denoted by $h^{\wedge}_{\pm}A$. Bhanja–K…