Antichain codes

A family of sets A$A$ is said to be an antichain if x & NSUB;y$x\not\subset y$ for all distinct x,y & ISIN;A$x,y\in A$, and it is said to be a distance-r$r$ code if every pair of distinct elements of A$A$ has Hamming distance at least r$r$. Here, we prove that if A & SUB;2[n]$A\subset 2<^>{[n]}$ is both an antichain and a distance-r$r$ code, then |A|=Or(2nn-1/2-L(r-1)/2<SIC> RIGHT FLOOR)$|A| = O_r(2<^>n n<^>{-1/2 - \lfloor (r-1)/2\rfloor } )$. 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 Halasz's theorem, while all previously known proofs of this result are Fourier-analytic.