Applying extremal graph theory to a question on finite groups

A word w∈ Fr is said to satisfy a probability gap if there exists a constant δw< 1 such that, for any finite group G, if the probability that w(g1, g2, … , gr) = 1 in G is at least δw, then w is an identity in G. Moreover we saythat a group G has the wt-property if however r subsets X1,.. , Xr of G are chosen with | X1| = ⋯ = | Xr| = t, there exists (g1, … , gr) ∈ X1× ⋯ × Xr such that w(g1, … , gr) = 1. We prove that if w satisfies a probability gap, then for every positive integer t there exists a constant ct such thatif a finite group G satisfies the wt-property, then | G| ≤ ct or w is an identity in G.