If $\mathcal{A}\subset\mathbb{N}$ is such that it does not contain a subset $S$ consisting of $k$ pairwise coprime integers, then we say that $\mathcal{A}$ has the property $P_k$. Let $\Gamma_k$ denote the family of those subsets of $\mathbb{N}$ which have the property $P_k$. If $F_k(n)=\max_{\mathcal{A}\subset\{1,2,3,\ldots,n\},\mathcal{A}\in\Gamma_k}\vert\mathcal{A}\vert$ and $\Psi_k(n)$ is the number of integers $u\in\{1,2,3,\ldots,n\}$ which are multiples of at least one of the first $k$ primes, it was conjectured that $F_k(n)=\Psi_{k-1}(n)$ for all $k\geq2$. In this paper, we give several partial answers.
We prove in this paper that for $y=x^{\theta}, \frac{11}{20}<\theta\leq\frac{7}{12}$ and $x$ large enough, we have $0.99\frac{y}{\log x}\leq\pi(x)-\pi(x-y)\leq1.01\frac{y}{\log x}$.