Form equivalence class Howard-Rubin Number: 99

Statement:

Strong Rado Lemma 2: Let \(T\) be a family of finite sets and let \({\cal F}\) be a collection of finite subsets of \(T\) satisfying

  1. \((\forall K\in T)(\exists S\in {\cal F})(K\in S)\)
  2. \((\forall S_{1}, S_{2}\in{\cal F})(S_{1}\cup S_{2}\in {\cal F})\)
Assume there is a function \(\phi\) with domain \({\cal F}\) such that for 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 a function \(f\) 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-Rubin number: 99 C

Citations (articles): Howard [1993] Variations of Rado's lemma

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