We give here a new proof of Erdos-Rado canonizing theorem for pairs which yields the upper bound 2^{(3/2).m^6} in the finite case. The proof evolved naturally from a new proof of Ramsey theorem for pairs which we present too.

Remark:

H. Lefmann and V. Rodl proved much better upper bound 2^{c.m^2.log m} (Combinatorica 15 (1995), 85-104; see also J. Comb. Theory, Ser. B 58 (1993), 1-13).  For the best upper bound for k-tuples see S. Shelah (Comment. Math. Univ. Carol. 37 (1996), 445-456).