Antichain Codes
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A family of sets A is said to be an antichain if x⊄y for all distinct x,y∈ A, and it is said to be a distance-r code if every pair of distinct elements of A has Hamming distance at least r. Here, we prove that if A⊂ 2^[n] is both an antichain and a distance-(2r+1) code, then |A| = O_r(2^n n^-r-1/2). 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.
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