14 problems

Erdős–Turán conjecture. If $A$ is an asymptotic basis, then the representation function $f_A(n)$ cannot be bounded.

For a translatable semigroup $T$, let $S_T(h)$ be the minimum integer $s$ such that every basis $A$ with ${\mathrm{ord}}_T^*(A)\leq h$ has only finitely many elements $a\in A$ for…

Bukh's open problem. Do the same bounds as in the rational vector model hold over $\mathbb{R}^{n}$; in particular, does the corresponding sharp lower bound persist for every basis…

Bukh–van Hintum–Keevash's conjecture. Then

Let $k\in\mathbb{N}$ and let $A\subseteq\mathbb{N}\cap kB$ for some $B\subset\mathbb{Z}$ with $|B|=n$. Here $kB$ denotes the set of sums of $k$ elements of $B$. Near-linear domain-…

Let $B0,B1subset ewcommand{mathbb{Q}}{mathbb{Q}}mathbb{Q}^n$ be such that $B0+B1$ contains every vector $ei+ej$ with $1leq ileq jleq n$. Vector-model lower-bound conjecture. One sh…

Attainment conjecture. For all sufficiently large $m$,

Size conjecture for bounded-representation bases.

Mean representation conjecture.

Chen's conjecture. There are infinitely many positive integers $m$ such that

Lipschitz conjecture for Ruzsa's number. For every positive integer $m$,

Let $\mathbb{P}$ be the set of palindromic natural numbers, and let $n$ be a sufficiently large natural number. The largest and second largest elements of $\mathbb{P}$ not exceedin…

Let $A$ be an additive basis of order $h={\rm G}(A)$, and let $x\in A$ be such that $A\setminus\{x\}$ is still an additive basis. Here ${\rm G}(B)$ denotes the least order of an ad…

Let $\{t_n\}$ be the nondecreasing union, with 1 repeated twice, of the sequences $\{s_n(1,1)\}$ and $\{s_n(1,2)\}$. Let $T(x)$ count the terms $t_n\leq x$. Minimality conjecture.…