Abstract

We provide a short proof that if κ is a regular cardinal with κ ≤ c, then $\left( \matrix \kappa \\ \omega\endmatrix \right)\rightarrow \left( \matrix\format\c\kern.8em&\c\\ \kappa & \alpha \\ \omega & \omega \endmatrix \right)^{1,1}$ for any ordinal α < min {p, κ}. In particular, $\left( \matrix \germ{p} \\ \omega \endmatrix \right)\rightarrow \left( \matrix\format\c\kern.8em&\c\\ \germ{p} & \alpha \\ \omega & \omega \endmatrix \right)^{1,1}$ for any ordinal α < p. This generalizes an unpublished results of E. Szemerédi that Martin's axiom implies that $\left( \matrix \germ{c} \\ \omega \endmatrix \right)\rightarrow \left( \matrix\format\c\kern.8em&\c\\ \germ{c} & \kappa \\ \omega & \omega \endmatrix \right)^{1,1}$ for any cardinal κ with κ < c