Let $X$ be an infinite cset and let $S$ be a collection of subsets of $X$ such that for $s\neq t\in S$ we have $|s\cap t| \leq 1$.

Is it consistent with $\mathsf{ZF}$ (without $\mathsf{AC}$) that there is a injection $\varphi:\mathcal{P}(X)\to S$?