On regressive Ramsey numbers

We prove the following relation between regressive and classical Ramsey numbers R-reg(n) (n + 2) Rn-1(n) + 2. This is used to compute R-reg(3) (5) = 8, R-reg(4) (6) = 15, and R-reg(5)(7) greater than or equal to 36. We prove that R-x+k(2) (3 + k) - (k + 1), and use this to compute R-reg(2) (5) = 15. Finally, we provide the bounds 195less than or equal toR(reg)(2) (6) less than or equal to 5.2(42) + 2(39) - 2. (C) 2002 Elsevier Science (USA).