Minimum overlap problem

For a positive integer $n$, consider all ways of splitting $\{1,2,\dots,2n\}$ into two complementary sets $A=\{a_1,\dots,a_n\}$ and $B=\{b_1,\dots,b_n\}$, i.e. $A\cap B=\varnothing$, $A\cup B=\{1,2,\dots,2n\}$, and $|A|=|B|=n$. For such a splitting and an integer $k$ with $-2n\le k\le 2n$, let

$$M_k(A,B):=\#\{(i,j)\in\{1,\dots,n\}^2:\ a_i-b_j=k\}$$

be the number of representations of $k$ as a difference of an element of $A$ and an element of $B$, and put

$$M(n):=\min_{A,B}\ \max_{-2n\le k\le 2n} M_k(A,B),$$

the minimum taken over all such splittings of $\{1,2,\dots,2n\}$.

Determine the asymptotic size of $M(n)$ as $n\to\infty$: exhibit a constant $c$ and prove that $M(n)=cn+o(n)$, i.e. determine

$$\liminf_{n\to\infty}\frac{M(n)}{n}\qquad\text{and}\qquad \limsup_{n\to\infty}\frac{M(n)}{n},$$

and show that these two quantities coincide.