Abstract
The Ramsey–Schur number RS ( s , m ) is the smallest n such that every 2-coloring (green/red) of the edges of complete graph K n with vertices 1 , 2 , … , n contains a green complete subgraph K s or there are vertices x 0 , x 1 , x 2 , … , x m fulfilling the equation x 1 + x 2 + ⋯ + x m = x 0 and all edges ( x i , x j ) are red. In this paper we prove RS ( 3 , m ) = m 2 + 2 m − 2 for m ⩾ 3 , which confirms a conjecture of J. Bode, H.O.F. Gronau and H. Harborth.
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