Moss–Pedersen conjecture on balanced subfamilies of maximum complement-free families

Let $k\geq 1$ be an integer. A family $\mathcal F\subseteq\binom{[2n]}{n}$ is balanced if every element of $[2n]$ belongs to the same number of members of $\mathcal F$; a balanced subfamily of size $2$ is a complementary pair.

Moss–Pedersen conjecture. For all sufficiently large $n$, there exists a family $\mathcal F\subseteq\binom{[2n]}{n}$ such that

$$|\mathcal F|=\frac{1}{2}\binom{2n}{n},$$

$\mathcal F$ contains no balanced subfamily of size $2,4,\ldots,2k$, and $\mathcal F$ contains a balanced subfamily of size $2k+2$.

The first condition makes $\mathcal F$ as large as possible among families containing no complementary pair. The paper’s abstract states that the result is proved constructively, so the conjecture is resolved by the work presented here.