A non-implication between fragments of Martin's Axiom related to a property which comes from Aronszajn trees

We introduce a property of forcing notions, called the anti-R1,א1 , which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property R1,א1 . In this paper, we investigate the property R1,א1 . For example, we show that a forcing notion with the property R1,א1 does not add random reals. We prove that it is consistent that every forcing notion with the property R1,א1 has precaliber א1 and MAא1 for forcing notions with the property R1,א1 fails. This negatively answers a part of one of classical problems about implications between fragments of MAא1 .