Form equivalence class Howard-Rubin Number: 14
Statement: Strong Rado Lemma 1: Let \(T\) be a family of finitesets and let \({\Cal F}\) be a collection of finite subsets of \(T\)satisfying\itemitem{(1)} \((\forall K\in T)(\exists S\in{\Cal F})(K\in S)\)\itemitem{(2)} \((\forall S_{1}, S_{2}\in{\Cal F})(\exists S_{3}\in{\Cal F})(S_{1}\cup S_{2}\subseteq S_{3})\)\item{}Assume there is a function \(\phi\) with domain \({\Cal F}\) such thatfor each \(S\in{\Cal F}\), \(\phi(S) = \phi_{S}\) is a function with domain\(S\) and satisfies \((\forall K\in S)(\phi_{S}(K)\in K)\). Then there is afunction \(f\) with domain \(T\) and with the following property: For all\(S\in{\Cal F}\) there is an \(S'\in{\Cal F}\) such that \(S\subseteq S'\) and\(f\) and \(\phi_{S'}\) agree on \(S\). Howard [1993].
Howard-Rubin number: 14 AQ
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