The Daykin–Frankl conjecture on convex subsets of the Boolean lattice

Let $P\subseteq Q_n$ be convex. Williams proves the stronger statement that for every $k\ge0$, $$w(P\times Q_k) \ge w(Q_{n+k})\,|P|\,2^{-n},$$ where $w$ denotes poset width. Taking $k=0$ gives $$w(P)\ge |P|\binom{n}{\lfloor n/2\rfloor}2^{-n},$$ which is exactly the Daykin-Frankl conjecture. The proof proceeds by induction on $n$, reducing the step to a structural lemma for a convex subset of $R\times Q_1$ and carefully recombining antichains from its two layers.

Let $P\subseteq Q_n$ be convex. Williams proves the stronger statement that for every $k\ge0$, $$w(P\times Q_k) \ge w(Q_{n+k})\,|P|\,2^{-n},$$ where $w$ denotes poset width. Taking $k=0$ gives $$w(P)\ge |P|\binom{n}{\lfloor n/2\rfloor}2^{-n},$$ which is exactly the Daykin-Frankl conjecture. The proof proceeds by induction on $n$, reducing the step to a structural lemma for a convex subset of $R\times Q_1$ and carefully recombining antichains from its two layers.