18 problems

Let $A\subseteq\{1,2,\ldots,n\}$ have distinct subset sums (SSD), meaning that all sums of subsets of $A$ are distinct. The problem asks for the maximum possible cardinality of $A$…

Folkman's conjecture. There is a constant $c$ such that every increasing sequence satisfying

Erdős's conjecture. There is a constant $c$ such that every increasing sequence satisfying

Folkman's conjecture. There is a constant $C$ such that, if $A(n)\geq Cn$ for all sufficiently large $n$, then $A$ is subcomplete.

Folkman's conjecture. A sufficiently dense sequence of positive integers, with possible repetitions, is subcomplete.

Growth-rate conjecture. We have

Equidistribution conjecture. The subset sums of $h$-element subsets of the cyclic group $\mathbb{Z}_n$ should satisfy

Conway–Guy conjecture. All sets arising from the Conway–Guy construction have distinct subset sums and are close to the best possible with respect to their largest element.

Erdős's conjecture. There is a constant $c$ such that

Let $t\geq 0$ be fixed and let $n$ be sufficiently large. Set $N=2^n+t$, and let $A$ be an $n$-element set of positive integers whose subset sums are all distinct modulo $N$. Modul…

Let $\mathcal{W}=\{w_1,w_2,\ldots,w_n\}$ be a set of weights with all distinct subset sums, meaning that the subset sums of $\mathcal{W}$ are pairwise distinct. Let $w$ denote the…

Alon's conjecture. In roughly the range $n^{1+\epsilon}\leq m\leq n^2/(\log n)^2$, one has

Conjecture on the optimal bound. The example consisting of groups of repeated coordinates is essentially the worst possible, so that

Erdős's conjecture. There is a constant $c>0$ such that

Let $z\in\mathbb{N}$ and let $A=\{a_1,\ldots,a_k\}\subseteq[2^z]$, where $[2^z]=\{1,2,\ldots,2^z\}$. For each non-empty subset $A_i$ of $A$, write $S(A_i)$ for the sum of its eleme…

Let $A$ be a finite set of $n$ positive integers, let $d_k(A)$ be the number of $k$-subsets $B$ of $A$ satisfying $\sum B\mid\sum A$, and let $d(k,n)$ be the maximum of $d_k(A)$ ov…

Anti-pencil conjecture. If $n\geq 4k$, then

Let $G$ be a finite abelian group, and let $S\subseteq G\setminus\{0\}$ be a subset. For a subset $S$ of an abelian group, write $\Sigma(S)$ for its set of subset sums, and say tha…