Two-level Littlewood-Offord bound ($\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)+\left(\genfrac{}{}{0ex}{}{n}{\lfloor (n-1)/2\rfloor }\right)$)

For real coefficients of absolute value at least one, an open interval of length four contains at most the sum of the two largest binomial coefficients of signed sums. The corresponding subset family has no three-element inclusion chain. Its Lubell mass is therefore at most two, giving the bound by allocating mass to the two largest levels. Unit coefficients attain it at a center midway between the two chosen sum levels.