A measure of thickness for families of sets (Q1069936)

For a set S, P(S) is the system of non-negative \(\lambda\) functions from S with \(\sum \{\lambda (s):\) \(s\in S\}=1\), and \(\{\) \(s\in S:\) \(\lambda (s)>0\}\) finite. If F is a family of subsets of S, then \(e(F)=\inf \sup \lambda (W)\), where the sup is taken for \(W\in F\), the inf is taken for \(\lambda\in P(S)\). It is shown that for every \(\alpha\) between 0 and 1 there are S, F with \(e(F)=\alpha\). There are S, F with \(e(F)=0\), but the inf is positive if we restrict to those \(\lambda\) with \(\lambda (x)=\lambda (y)\) if \(\lambda\) (x), \(\lambda (y)>0\). It is described when it is possible to find R, an infinite subset of S, such that the sup is positive if we restrict to those \(\lambda\) which support a subset of R.