Notes

Description: In Zuckerman [1969a] and Zuckerman [1981], the relationship between versions of Form 178 (\(n,N\)) are considered.

Content:

In Zuckerman [1969a] and Zuckerman [1981], the relationship between versions of Form 178 (\(n,N\)) are considered. (For reference, Form 178 (\(n,N\)) is:

If \(n\in\omega\), \(n\ge 2\) and \(N\subseteq \{1, 2,\ldots, n-1\}\),\(N\neq\emptyset\), \(MC(\infty,n,N)\): If \(X\) is any set of \(n\)-element sets then  there is a function \(f\) with domain \(X\) such that \(\forall A \in X\), \(f(A)\subseteq A\) and \(|f(A)|\in N\).)

A sample of the results follows. Using the notation of Zuckerman [1981], \(S(n,\ell)\) is \(MC(\infty,n, \{\ell\})\) and \(T(n,\ell)\) is \(MC(\infty,n,\{1,2,\ldots,\ell\})\).

Theorem:  If \(m\le n\) and \(n-\ell \le m\) then \(S(n,\ell)\rightarrow  T(m,\ell)\).

Theorem: If \(l < n\) and \(k \ge  0\), then \(S(n,\ell )\wedge  S(kn + \ell ,1)\)  imply \(S(n,1)\).

Theorem:  Assume \(K \ge  1 \)

  1. \(\ell < n\), \(T(kn,\ell ) \rightarrow  T(n,\ell )\).
  2. for \(\ell \) not of the form \(jn\) for any \(j \le  k\), \(S(kn,\ell ) \rightarrow  T(n,l)\).

Theorem:  If \(l < n\) and \(k \ge  0\), then \(S(n,\ell )\wedge  S(kn + \ell ,1)\)  imply \(S(n,1)\).

Assume \(K \ge  1 \)

  1. \(\ell < n\), \(T(kn,\ell ) \rightarrow  T(n,\ell )\).
  2. for \(\ell \) not of the form \(jn\) for any \(j \le  k\), \(S(kn,\ell ) \rightarrow  T(n,l)\).

Howard-Rubin number: 53

Type: Summary of theorems

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