8 problems

Let $\mathbb{N}=\bigcup_{i=1}^m A_i$ be a finite partition, let $\mathcal{P}_f(\mathbb{N})$ denote the nonempty finite subsets of $\mathbb{N}$, and let $\alpha<\beta$ mean…

Let a finite partition of $\mathbb{N}$ be given. Hindman's question. There exist $x,y\in\mathbb{N}$ such that … is monochromatic. This asks whether every finite coloring of the pos…

Entropy sum lower-bound conjecture. The constant $1/8$ can be improved to $1/2$, meaning that

Small-representative conjecture. If $(i,j)\in\operatorname{SPP}(n)$, then there is a set $A\subseteq\mathbb N$ with

Let $\operatorname{NSPP}([3,\infty))$ denote the multiset of normalized sum-product pairs for positive-integer sets of cardinalities at least $3$, with each pair lying in $[1,2]^2$…

Sum-product exponent conjecture. Then

Solymosi–Volkmann conjecture. One should have

Erdős's sum-product conjecture. For every $\epsilon>0$ there is an $n_0$ such that, whenever $|A|\ge n_0$,