Diamond and antichains (Q1765105)

Let \(\langle s_\alpha: \alpha< \omega_1\rangle\) be a \(\diamondsuit\)-sequence. For \(a\subseteq\omega_1\), let \(T_a= \{\xi< \omega_1: s_\xi= a\cap\xi\}\). Given \(\alpha< \omega_1\), define by recursion \(s^\delta_\alpha\subseteq \alpha+1\) for \(\delta< \omega_2\) by letting \(s^\delta_\alpha\) be \(\{\beta\leq \alpha:\sup(s_\alpha\cap \beta)= \beta\}\) if \(\delta= 0\), \(\{\beta\leq\alpha: \text{o.t.}(\beta\cap T_{s_\alpha}\cap s^\gamma_\alpha)= \beta\}\) if \(\delta= \gamma+1\), \(\bigcap_{\gamma<\delta} s^\gamma_\alpha\) if \(\delta\) is a countable limit ordinal, and \(\{\beta\leq \alpha:\exists \eta<\beta\) \((\beta\in s^{e_\delta(\eta)}_\alpha)\}\) if \(\delta\) is an uncountable limit ordinal, where \(e_\delta\) is a fixed function from \(\omega_1\) onto \(\delta\) with \(e_\delta(0)= 0\). For \((\delta,\rho)\in \omega_2\times \omega_1\), let \(T_{(\delta,\rho)}\) be the set of all \(\alpha< \omega_1\) such that \(T_{s_\alpha}\cap s^\delta_\alpha\) has order type \(\rho+ 1\) and greatest element \(\alpha\). It is shown that \(T_x\cap T_y\) is a bounded subset of \(\omega_1\) for any two distinct members \(x\), \(y\) of the set \(P(\omega_1)\cup (\omega_2\times \omega_1)\).