It is known that for all monotone functions f : {0, 1}n → {0, 1}, if x ∈ {0, 1}n is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ϵ = n−α, then P[f(x) ≠ f(y)] < cn−α+1/2, for some c > 0. Previously, the best construction of monotone functions satisfying P[fn(x) ≠ fn(y)] ≥ δ, where 0 < δ < 1/2, required ϵ ≥ c(δ)n−α, where α = 1 − ln 2/ln 3 = 0.36907 …, and c(δ) > 0. We improve this result by achieving for every 0 < δ < 1/2, P[fn(x) ≠ fn(y)] ≥ δ, with: ϵ = c(δ)n−α for any α < 1/2, using the recursive majority function with arity k = k(α); ϵ = c(δ)n−1/2logtn for t = log2 = .3257 …, using an explicit recursive majority function with increasing arities; and ϵ = c(δ)n−1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand. We also study the problem of achieving the best dependence on δ in the case that the noise rate ϵ is at least a small constant; the results we obtain are tight to within logarithmic factors.