Well-spread sequences and edge-labellings with constant Hamilton-weight

A sequence (a_i) of integers is well-spread if the sums a _i + a_j, for i < j, are all different. For a fixed positive integer r, let W_r(N) denote the maximum integer n for which there exists a well-spread sequence 0 ≤ a₁ < ⋯ < a_n ≤ N with a_i ≡ a_j (mod r) for all i, j. We give a new proof that W_r(N) < (N/r)^1/2 + O((N/r)^1/4); our approach improves a bound of Ruzsa [Acta. Arith. 65 (1993), 259-283] by decreasing the implicit constant, essentially from 4 to √3. We apply this result to verify a conjecture of Jones et al. from [Discuss. Math. Graph Theory 23 (2003), 287-307]. The application concerns the growth-rate of the maximum label Λ(n) in a 'most-efficient' metric, injective edge-labelling of K_n with the property that every Hamilton cycle has the same length; we prove that 2n² - O(n^3/2) < Λ(n) < 2n² + O(n^61/40).