Motivated by questions concerning the multilinear and homogeneous complexity of the elementary symmetric polynomials, we prove the following results:

We first show that by making small modifications to the nonzero coefficients of the degree-$K$, $N$-variate elementary symmetric polynomial $\sigma_{N,K}$, one obtains a polynomial that can be computed by a monotone formula of size $K^{O(\log K)} \cdot N$.

As a corollary, we show that a result of Raz [Raz13] concerning the homogenization of algebraic multilinear or monotone formulas is tight.

Another corollary is that the monotone bounded rigidity of the inclusion matrix between $K$-subsets and $N-K$ subsets of a universe of size $N$ is small.