4 problems

Let $A$ be a finite subset of a ring. For an integer $k\geq2$, let $kA$ denote the $k$-fold sum set and let $A^{(k)}$ denote the $k$-fold product set. Erdős's many sums and product…

Sum-product conjecture. In a given ring,

Let $A\subset\mathbb{R}$ be a finite set. If $\xi\in\mathbb{R}$ is transcendental and $K_1,K_2\geqslant 2$ satisfy … then The real polynomial Freiman-Ruzsa conjecture. … This conje…

Finite-VC-dimension field conjecture. The set $F$ constitutes an uncountable subfield of $\mathbb{R}$, analogous to the constructable numbers or computable numbers.