OAI.Problem355.almost_n_minus_two_refuted

The theorem states that, with ε* = heilbronnExponent = 1/(100000·K), where K = T²+1, T = C(M,3) and M = C(4·41−1, 41) = C(163,41) (a fixed explicit positive constant), the number ε*/2 is positive, and the following eventual almost-upper-bound property fails for ε = ε*/2. That property, eventualAlmostUpperBound(ε), asserts the existence of a real C>0 and a natural number n₀ such that for every n ≥ n₀ with n ≥ 3 and every set P of exactly n points of the unit square [0,1]×[0,1], some three distinct points of P span a triangle of area at most C·n^(−2+ε). So the theorem asserts that no such constant C and threshold n₀ exist for ε = ε*/2, that is, for every C>0 and n₀ there are some n ≥ max(n₀,3) and an n-point subset of the unit square in which all triangles formed by distinct points have area greater than C·n^(−2+ε*/2). The statement is admitted in the source rather than proved here.