Generalized ramsey theory for graphs, x: double stars

The double star S(n, m), where n * m a 0, is the graph cons isting of the union of two stars &I and &n together with a line joining their centers. Its rsmsey number r(S(n, m)) is the least number p such that there is a monochromatic copy of S(n, m) in any 2-coloring of the edges of Kpa It is shown that r(S(n, m)) = max (2n + 1, n + 2m + 2) if n is odd and m G 2; and r(S(n, m)) = max (2n + 2, n + 2m + 2) otherwise, for n ~&z or n 3 3m.