An exact computer-assisted certificate for R(5,6)≤84

We give an exact computer-assisted proof of R(5,6)≤84. We derive a system of linear inequalities that must be satisfied by any graph on 84 vertices containing neither a five-vertex clique nor a six-vertex independent set. We provide explicit nonnegative rational weights for which adding these inequalities yields a contradiction. Published small Ramsey bounds and complete catalogues of smaller Ramsey graphs supply the external inputs. The accompanying verification programs check the derivation of the inequalities and their weighted sum using exact arithmetic. Two supplements give a worked local calculation and a detailed mathematical specification of the complete proof.