Consistency of square bracket partition relation

Abstract

Characteristic earlier results were of the form CON$(2^{\aleph_0} \to [\lambda]^2_{n, 2})$, with $2^{\aleph_0} $ an ex-large cardinal, in the best case the first weakly Mahlo cardinal. Characteristic new results are CON$((2^{\aleph_0} = \aleph_m) + \aleph_l \to [\aleph_k]^2_{n, 2})$, for suitable $k < l < m$. So we improve in three respects: the continuum may be small (e.g. not a weakly Mahlo), we use no large cardinal, and the cardinals $\lambda$ involved are $ < 2^{\aleph_0}$ after the forcing.