A generalization of Martin’s Axiom

We define the ℵ1.5-chain condition. The corresponding forcing axiom is a generalization of Martin’s Axiom; in fact, \({\text{MA}}_{ < \kappa }^{1.5}\) implies \({\text{M}}{{\text{A}}_{ < \kappa }}\). Also, \({\text{MA}}_{ < \kappa }^{1.5}\) implies certain uniform failures of club-guessing on ω 1 that do not seem to have been considered in the literature before. We show, assuming CH and given any regular cardinal κ ≥ ω 2 such that \({\mu ^{{\aleph _0}}} < \kappa \) for all µ < κ and such that ⋄({α < κ: cf(α) ≥ω 1}) holds, that there is a proper ℵ2-c.c. partial order of size κ forcing \({2^{{\aleph _0}}} = \kappa \) together with MA <κ 1.5 .