{\mathcal F}_1 = \{ \emptyset\}, {\mathcal F}_2 = \{ \{1\}\}, {\mathcal F}_3 = \{\{1,2\}\}, {\mathcal F}_4 = \{\{1,2,3\}\}, {\mathcal F}_5 = \{\{2,3\}\}, {\mathcal F}_6 = \{\{3\}\};

optimal?

Why cant I shove in {{2}} to get

{\mathcal F}_1 = \{ \emptyset\}, {\mathcal F}_2 = \{ \{1\}\}, {\mathcal F}_3 = \{\{2\}\}, {\mathcal F}_4 = \{\{1,2\}\}, {\mathcal F}_5 = \{\{1,2,3\}\}, {\mathcal F}_6 = \{\{2,3\}\}, {\mathcal F}_7 = \{\{3\}\};

(I may be making a silly error, but I cannot see it)

]]>*[Corrected, thanks – T.]*

*[Ah, I see the problem now: F_6 had run off beyond the column width of the blog and so was not visible. Fixed now – T.]*