Grid Ramsey problem

The grid Ramsey number G(r) is the smallest number n such that every edgecolouring of the grid graph Γn,n:=Kn×Kn with r colours induces a rectangle whose parallel edges receive the same colour. We show G(r)≤r(r+12)−(1/4−o(1))r(r2)+1 slightly improving the currently best known upper bound due to Gyárfás.