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- R 1 , ℵ 1 , which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property R 1 , ℵ 1 . s paper, we investigate the property R 1 , ℵ 1 . For example, we show that a forcing notion with the property R 1 , ℵ 1 does not add random reals. We prove that it is consistent that every forcing notion with the property R 1 , ℵ 1 has precaliber ℵ 1 and MA ℵ 1 for forcing notions with the property R 1 , ℵ 1 fails. This negatively answers a part of one of the classical problems about implications between fragments of MA ℵ 1 .