Square principle

For an infinite cardinal $\kappa$, the square principle supplies club sets $C_{\zeta }\subseteq \zeta$ at limit $\zeta <{\kappa }^{+}$, with order type at most $\kappa$, coherent at limit points: $C_{\gamma }=C_{\zeta }\cap \gamma$ whenever $\gamma$ is a limit point of $C_{\zeta }$. The bound precludes a single club set threading the entire sequence. For regular uncountable $\kappa$, coherence together with the weaker-looking size clause $\mathrm{cf}(\zeta )<\kappa \Rightarrow |C_{\zeta }|<\kappa$ implies the order-type bound. At $\kappa =\omega$, that clause alone is vacuous and cannot replace the bound.