10 problems

For a positive integer $n$, let $s(n,3)$ be the largest cardinality of a subset $S\subseteq\mathbb Z/n\mathbb Z$ such that every congruence … with $\varepsilon_i\in\{0,\pm1\}$ and…

Let $A$ be a $B_r$ sequence, meaning that every integer has at most one representation as a sum of $r$ elements of $A$, with summands ordered nondecreasingly, and let $A(n)$ count…

Erdős's logarithmic density conjecture. For every $h\geq2$,

Let $A$ be a $B_6$ sequence and let $A(x)=\#(A\cap[0,x])$. Unrestricted $B_6$ density conjecture. The condition $A(x^2)={\cal O}(A(x)^2)$ can be omitted from the corresponding dens…

Let ${\cal G}_2[2]=\{\gamma_1,\gamma_2,\ldots\}$ be the greedy Sidon sequence, with successive terms chosen as the least possible integers preserving the Sidon property. Erdős's su…

A $B_2[k]$ sequence is a sequence in which every integer has at most $k$ representations as a sum of two elements, up to rearrangement. Erdős–Newman conjecture. There exists a…

Consider the sequence of fifth powers … A Sidon sequence is a sequence in which all sums of two elements are distinct up to rearrangement of the summands. Fifth-power Sidon conject…

Let ${\cal A}$ be a $B_h$ sequence, meaning that every integer has at most one representation as a sum of $h$ elements of ${\cal A}$, with summands counted up to rearrangement, and…

Let $h$ and $g$ be positive integers. A $B_h^*[g]$ sequence is a sequence in which every integer has at most $g$ representations as a sum of $h$ distinct elements, up to rearrangem…

Let $f_2(n)$ denote the maximum, over all $n$-vertex 2-connected graphs, of the number of cycle lengths that occur exactly once. The authors' conjecture is … This conjecture assert…