Constructing O(n log n) Size Monotone Formulae for the k-th Threshold Function of n Boolean Variables

In this paper we construct formulae for the kth elementary symmetric polynomial of n Boolean variables, using only conjunction and disjunction, which for fixed k are of size $O(n\log n)$, with the construction taking time polynomial in n. We also prove theorems involving $n\log n \cdot $ (polynomial in k) upper bounds on such formulae. Our methods involve solving the following combinatorial problem: for fixed k and any n construct a collection of $r = O(\log n)$ functions $f_1 , \cdots ,f_r $ from $\{ 1, \cdots ,n\} $ to $\{ 1, \cdots ,k\} $ such that any subset of $\{ 1, \cdots ,n\} $ of order k is mapped 1–1 to $\{ 1, \cdots ,k\} $ by at least one $f_i $.