10 problems

Gap-free floor conjecture.

Let $A$ be an increasing sequence of positive integers, allowing repetitions, and let $A(n)$ denote the number of elements of $A$, counted with multiplicity, that are at most $n$.…

Let $A$ be a strictly increasing sequence of positive integers, and write $A(n):=|A\cap[n]|$, where $[n]:=\{1,2,\dots,n\}$. Thus $A$ has asymptotic density at least $C\sqrt n$ when…

Folkman'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.

Growth-rate conjecture. We have

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 $z\in\mathbb{N}$ 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