Notes

Description:

Form 234 implies Form 282

Content:

In this note we prove that Form 234 (There is a non-Ramsey set.) implies Form 282 (\(\omega\not\to(\omega)^\omega\)).   Assume that there is a non-Ramsey set \(A\). The proof will be by contradiction.  We therefore assume that \(\omega\to (\omega)^\omega\).  Then (using the definitions in Note 97) for every \(F:\omega^\omega \to \{0,1\}\) there is an infinite subset \(X\) of \(\omega\) such that either

  1. \((\forall g \in X^\omega)(F(g) = 0)\) or
  2. \((\forall g \in X^\omega)(F(g) = 1)\).
Using \(A\) we define a function \(F:\omega^\omega \to \{0,1\}\) by \[F(g) = \begin{cases} 0  & \hbox{if g is strictly increasing and range} (g) \in A \\ 1  & \hbox{otherwise} \\ \end{cases} \] Let \(X\) be the infinite subset of \(\omega\) guaranteed by\(\omega\to(\omega)^\omega\).  For each infinite subset \(Y\) of \(\omega\),let \(g_Y \in\omega^\omega\) be the unique strictly increasing function whose range is \(Y\). Then \(F(g_Y) = 0\) if and only if \(Y\in A\).  Either (i) or (ii) must be true of \(X\) and \(F\). If (i) holds then for every infinite \(Y\subseteq X\), \(g_Y\in X^\omega\) so \(F(g_Y) = 0\) and hence \(Y\in A\).  On the other hand, if (ii) holds then for every infinite \(Y\subseteq X\), \(g_Y\in X^\omega\) so \(F(g_Y) = 1\) and hence \(Y\notin A\).  This contradicts the assumption that \(A\) is non-Ramsey.

Howard-Rubin number: 142

Type: Proof

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