The lower bound of the number of inequalities which represent a monotonic boolean function of n variables

It is shown that there is a monotonic Boolean function whose representation by a system of linear inequalities with n Boolean variables requires not less than ( n [ n / 2 ] ) n − 1 inequalities. Suppose that we have the following system of non-linear inequalities: ∑ j − 1 n a i j x j ≤ b i , i = 1 , 2 , ... , m , (1) where the coefficients aij, bi i=1, 2, ... m, j=1, 2, ..., n are non-negative real numbers, and the variables xj, j=1, 2, ..., n are Boolean.