A combinatorial linear programming problem

Question

$\newcommand\S{\mathscr S}$Let $\S$ be a collection of nonempty subsets of a finite set $S$ such that $A\not\subset B$ for any distinct $A$ and $B$ in $\S$.

Does then there always exist a function $f\colon S\to(0,\infty)$ such that $\sum_{s\in A}f(s)=1$ for all $A\in\S$?

The answer to this question is yes if each set $A$ in $\S$ is of cardinality $\le2$.

Answer 1

There isn't even a solution if one drops the positivity requirement on $f$. Set \begin{equation} \mathcal S=\{\{0, 1\}, \{0, 2\}, \{0, 3\}, \{0, 4\}, \{1, 2\}, \{1, 3, 4\}\}. \end{equation} From the first $4$ members of $\mathcal S$ we get $f(1)=f(2)=f(3)=f(4)$. Together with the $5$th member we get $f(1)=f(2)=f(3)=f(4)=1/2$, so the last member yields the contradiction $1=f(1)+f(3)+f(4)=3/2$.