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A family of sets is said to be an antichain if for all distinct , and it is said to be a distance‐ code if every pair of distinct elements of has Hamming distance at least . Here, we prove that if is both an antichain and a distance‐ code, then . This result, which is best‐possible up to the implied constant, is a purely combinatorial strengthening of a number of results in Littlewood–Offord theory; for example, our result gives a short combinatorial proof of Hálasz's theorem, while all previously known proofs of this result are Fourier‐analytic.

We elucidate the relationship between the threshold and the expectation‐threshold of a down‐set. Qualitatively, our main result demonstrates that there exist down‐sets with polynomial gaps between their thresholds and expectation‐thresholds; in particular, the logarithmic gap predictions of Kahn–Kalai and Talagrand (recently proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about up‐sets do not apply to down‐sets. Quantitatively, we show that any collection of graphs on that covers the family of all triangle‐free graphs on satisfies the inequality for some universal , and this is essentially best‐possible.