Given any positive integers $n$ and $k$, consider the finite sequence of consecutive integers $n, n+1, \dotsc, n+k-1$, denoted by the interval $[n,n+k-1]$. We would like to find a maximal subset $\{a_1, a_2, \dotsc, a_t\}$ that is pairwise relatively prime. Define the maximum (not just maximal) size by $F(n,k)=\max t$. We are interested in the minimum value of $F(n,k)$ over $n$, i.e., $s(k)=\min_n F(n,k)$. In other words, we are interested in knowing that among any consecutive $k$ positive integers, how many pairwise coprime numbers we can guarantee to find.
In the paper "Complete prime subsets of consecutive integers" by P. Erdős and J. L. Selfridge on the Proceedings of the Manitoba Conference on Numerical Mathematics, Winnipeg (1971), it was found that $s(k)\geq k^{\frac{1}{2}-\epsilon}$. However, they also mention that the true value of $s(k)$ is probably closer to $\frac{k}{\log k}$. Our question is, is there a way to improve this bound to $k^{\frac{1}{2}}$? Is there any other literature that has worked on this problem?
Another related question. In the same finite sequence of consecutive integers $[n,n+k-1]$, we would like to find a minimal set of primes $\{p_1, p_2, \dotsc, p_\tau\}$ that "covers" the integer interval, i.e., every integer in $[n,n+k-1]$ is divisible by some $p_i$, where $1\leq i\leq\tau$. Define the minimum (not just minimal) size $g(n,k)=\min\tau$. We are interested in the minimum value of $g(n,k)$ over $n$, i.e., $t(k)=\min_n g(n,k)$. In other words, we are interested in knowing that among any consecutive $k$ positive integers, how many primes we must have in order to have a chance to cover the interval.
It is obvious that $t(k)\geq s(k)$. Again, our question is, is it true that $t(k)\geq k^{\frac{1}{2}}$?