4 problems

Let $g(\theta)$ be the least exponent such that, for some $c_1,c_2>0$, every interval of the form … contains an integer $n=a_1b_1=a_2b_2$, where … and all four integers lie in … an…

Let $f(\theta)$ be the least exponent such that, for some $c_1,c_2>0$, every interval of the form … contains an integer $n=ab$, where $a$ and $b$ lie in … and $0\leq\theta<1/2$; th…

Ruzsa's conjecture. There is a constant $K_\epsilon$ such that, for every positive integer $n$, the number of such divisors is at most $K_\epsilon$.

Erdős–Rosenfeld conjecture. There is an absolute constant $K$ such that, for every $c$, the number of such divisors is at most $K$ whenever $n>n_0(c)$.