176 problems

Erdős–Heilbronn conjecture. For every nonempty subset $A\subseteq {\mathbb Z}/p{\mathbb Z}$,

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…

Sárközy's conjecture. For all sufficiently large primes $p$, the quadratic residues modulo $p$ are additively irreducible.

Erdős's conjecture. For every such $A$, there exist an infinite $B\subseteq A$ and an integer $t$ such that

Let $n,k\in\mathbb{N}$, let $\lambda>0$, and let $\delta>0$. For a set $A\subseteq\{1,\ldots,n\}$, its doubling factor is measured by $|A+A|/|A|$. The construction underlying the c…

The asymptotic sum-dilate conjecture. $$

Exponential decay conjecture. There exists a constant $\lambda$ such that, for every $\epsilon>0$,

Threshold conjecture. For every fixed positive integer $d$, there should be a threshold around

Let ${\Cal B}\subset{\Bbb Z}$ have cardinality $N$, and write ${\Cal B}+{\Cal B}=\{b+b':b,b'\in{\Cal B}\}$. The estimate discussed in the source gives a lower bound of order…

Generalized Sárközy conjecture. Let $k\geq2$ be fixed. Then, for all sufficiently large prime powers $q\equiv1\pmod k$, the subgroup $A_k<\mathbb{F}_q^\times$ admits no nontrivial…

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…

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…

Let $k\in\mathbb{N}$ and let $\epsilon$ be positive. For finite sets $A,B\subset\mathbb{Z}^k$, say that $B$ is not covered by $n$ parallel hyperplanes when no collection of $n$ par…

Let $G$ be an abelian group, let $p$ be the smallest prime divisor of the order of $G$, and let $A$ be an $m$-subset of $G$. Assume that $m$ and $h$ are positive integers satisfyin…

Let $G$ be an abelian group, let $m$ be a positive integer, and let $L$ denote the subgroup of involutions in $G$. Write $\rho\hat{\;} (G,m,2)$ for the minimum size of a restricted…

Ruzsa's conjecture. If there exists a constant $K$ such that

Conway's conjecture. Every set $A\subset \mathbb Z$ satisfies

Let $k,T,d\bin \mathbb{N}$, and let $c,b$ satisfy … where … Here $\operatorname{vol}(k,T,d)$ is the maximum volume of a set of integers of cardinality $k$, doubling $T$, and additi…

Let $A$ be a subset of a finite-dimensional vector space over $\mathbb{F}_2$, and let the relevant quantities and bounds be those established earlier in the source for downsets, in…

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…

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…

Critical pair conjecture. Under these hypotheses, such a common difference $r$ exists.

The $(3k-4)$-conjecture. The conclusion that $A$ is contained in a short arithmetic progression should hold without the restriction that $|A|$ is small relative to $p$.