Bounded forcing axioms and the continuum

We show that bounded forcing axioms (for instance, the Bounded Proper Forcing Axiom and the Bounded Semiproper Forcing Axiom) are consistent with the existence of (!2; !2)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen’s combinatorial principles for L at the level !2, and therefore with the existence of an !2-Suslin tree. We also show that the axiom we call BMMא3 implies אא1 2 =א2, as well as a stationary re7ection principle which has many of the consequences of Martin’s Maximum for objects of size א2. Finally, we give an example of a so-called boldface bounded forcing axiom implying 2א0 = א2. c © 2001 Elsevier Science B.V. All rights reserved. MSC: 03E35; 03E50; 03E05; 03E65