Consistency of square bracket partition relation

TL;DR

This paper explores the consistency of certain partition relations involving smaller continuum sizes and without large cardinal assumptions, extending previous results that required large cardinals.

Contribution

It establishes new consistency results for partition relations with smaller continuum and no large cardinal assumptions, improving upon earlier work.

Findings

Shows consistency of partition relations with continuum as leph_m

Demonstrates relations hold for leph_k with smaller continuum

Eliminates need for large cardinal assumptions in these relations

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.