On a result of Szemerédi

Abstract

We provide a short proof that if $\kappa$ is a regular cardinal with $\kappa \leq \mathfrak{c}$, then $\left(\begin{matrix} \kappa \\ \omega \end{matrix}\right) \rightarrow \left(\begin{matrix} \kappa~~~\alpha \\ \omega~~~\omega \end{matrix} \right)^{1,1}$ for any ordinal $\alpha$ < min {$\mathfrak{p},\kappa$}. In particular, $\left(\begin {matrix} \mathfrak{p} \\ \omega \end{matrix}\right) \rightarrow \left(\begin{matrix} \mathfrak{p}~~~\alpha \\ \omega~~~\omega \end{matrix}\right)^{1,1}$ for any ordinal $\alpha < \mathfrak{p}$. This generalizes an unpublished result of E. Szemerédi that Martin’s axiom implies that $\left(\begin{matrix} \mathfrak{c} \\ \omega \end{matrix}\right) \rightarrow \left(\begin{matrix} \mathfrak{c}~~~\kappa \\ \omega~~~\omega \end{matrix}\right)^{1,1}$. for any cardinal $\kappa$ with $\kappa < \mathfrak{c}$.