The constructive diagonal Ramsey problem asks, given adjacency-oracle access to an N-vertex graph, for a clique or independent set of the order guaranteed by Ramsey's theorem. We give a bounded-error quantum algorithm that, for every K\ge2 and $N\
The constructive diagonal Ramsey problem asks, given adjacency-oracle access to an N-vertex graph, for a clique or independent set of the order guaranteed by Ramsey's theorem. We give a bounded-error quantum algorithm that, for every K\ge2 and N\ge4^{K-1}, finds and verifies a homogeneous K-set using O\!\left(2^K K\log\frac K\eta\right) edge queries with failure probability at most \eta. At the Ramsey scale N=2^n, this yields a homogeneous set of order \lfloor n/2\rfloor+1 using O(\sqrt N\log N\log(\log N/\eta)) queries, improving on the O(N) queries of the explicit classical recursion and giving, to our knowledge, the first sublinear worst-case algorithm for the Ramsey relation. We also derive an \Omega(N^{1/12}) quantum lower bound by a reduction from collision finding. The algorithm runs the constructive recursion over implicit candidate sets. Each set is represented by a short conjunction of adjacency constraints and sampled using capped unknown-solution quantum search, and a scale-aware concentration schedule balances estimation accuracy against the increasing cost of sampling deeper sets. We complement the upper bound with an \Omega(N^{1-1/\sqrt2}) randomized lower bound, transported from the random-Painter analysis of online Ramsey numbers, which holds on the uniform distribution G(N,1/2). On that distribution a greedy quantum search uses only \widetilde O(N^{1/4}) queries, giving a provable polynomial quantum speedup for Ramsey search on random graphs. We also give an estimation-free size-biased recursion and extend it to every fixed number of edge colours.