Maximum Cardinality of MSTD Subsets of Integer Intervals

For a finite set A, write $A+A$ and $A-A$ for its sumset and difference set, and call A MSTD if $|A+A|>|A-A|$. For n≥16, let Iₙ=\0,1,…,n-1\, and let Mₙ be the largest cardinality of an MSTD subset of Iₙ. Chu proved n-7≤ Mₙ≤ n-4 and conjectured that Mₙ=n-7. We prove this conjecture for every n≥16. The main step is a sharp endpoint-restricted theorem: every MSTD subset of Iₙ containing both endpoints omits at least seven elements, and this bound is attained. To prove the upper bound, we encode a missing positive difference d as a vertex-cover condition in the distance graph Gn,d, whose connected components are paths. An exact analysis of endpoint-avoiding minimum vertex covers, combined with Zhao's order-$5$ fringe-pair enumeration, excludes endpoint-containing MSTD sets with complements of size at most six. Known constructions provide the matching lower bound, and a reduction to the smallest containing interval yields the unrestricted result. The proof is uniform and avoids the finite computations previously used to verify the conjecture for small values of n.