Asymptotic Erdős Matching Conjecture

Fix an integer $k\geq2$ and a positive real number $x\leq1/k$. Let $m(n,k,xn)$ denote the maximum size of a $k$-uniform family on $[n]$ whose matching number is less than $xn+1$. Asymptotic Erdős Matching Conjecture. One has

\lim_{n\to\infty}\frac{m(n,k,xn)}{{n\choose k}}=\max\left\\{1-(1-x)^k,(kx)^k\right\\}.

This is the linear-density asymptotic relaxation of the Erdős Matching Conjecture, comparing the limiting densities of its two natural constructions. The source supplies no resolution status.