Hint: Expand upon the “random chain” argument from the proof of the LYM Inequality.
Prove that, for any \(k\geq0\) and \(\mathcal{A}\subseteq 2^{[n]}\) with \(|\mathcal{A}|= \binom{n}{\lfloor n/2\rfloor} + \binom{n}{\lfloor (n+2)/2\rfloor}+k\text{,}\) the number of pairs \(A,B\in \mathcal{A}\) such that \(A\subsetneq B\) is \(\Omega(k\cdot n^2)\text{.}\)
Prove that there exists a constant \(C>0\) such that the number of antichains in \(2^{[n]}\) is at most \(C^{\binom{n}{\lfloor n/2\rfloor}}\) for every \(n\text{.}\)
For \(0\leq k\leq n-1\text{,}\) let \(\mathcal{A}\subseteq \binom{[n]}{k}\) and \(\mathcal{B}\subseteq \binom{[n]}{k+1}\) be initial segments of colex order with \(|\mathcal{A}|=|\mathcal{B}|\text{.}\) Does it always hold that \(|\partial\mathcal{A}|\leq |\partial\mathcal{B}|\text{?}\)
