Question 12
(GDR 3) Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an S-family over $R$ if and only if it satisfies the following three conditions:
(i) For no two sets $X,Y$ in $F$ is $X\subseteq Y$;
(ii) For any three sets $X,Y,Z$ in $F$, $X\cup Y\cup Z\ne R$;
(iii) $\bigcup_{X\in F}X=R$.
We define $|F|$ to be the number of elements of $F$ (i.e., the number of subsets of $R$ belonging to $F$). Determine, if it exists, $h=\max|F|$, the maximum being taken over all S-families over $R$.
(i) For no two sets $X,Y$ in $F$ is $X\subseteq Y$;
(ii) For any three sets $X,Y,Z$ in $F$, $X\cup Y\cup Z\ne R$;
(iii) $\bigcup_{X\in F}X=R$.
We define $|F|$ to be the number of elements of $F$ (i.e., the number of subsets of $R$ belonging to $F$). Determine, if it exists, $h=\max|F|$, the maximum being taken over all S-families over $R$.