The exact values of two-color Ramsey–Schur numbers RS(3,m)

The Ramsey–Schur number RS(s,m) is the smallest n such that every 2-coloring (green/red) of the edges of complete graph Kn with vertices 1,2,…,n contains a green complete subgraph Ks or there are vertices x0,x1,x2,…,xm fulfilling the equation x1+x2+⋯+xm=x0 and all edges (xi,xj) are red. In this paper we prove RS(3,m)=m2+2m−2 for m⩾3, which confirms a conjecture of J. Bode, H.O.F. Gronau and H. Harborth.