OFFSET
0,3
COMMENTS
The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
Excludes all set-systems with more edges than covered vertices, but this condition is not sufficient.
LINKS
Wikipedia, Axiom of choice.
EXAMPLE
The a(3) = 15 set-systems:
{}
{{1,2}}
{{1,3}}
{{2,3}}
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,2},{1,2,3}}
{{1,3},{2,3}}
{{1,3},{1,2,3}}
{{2,3},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,2},{1,3},{1,2,3}}
{{1,2},{2,3},{1,2,3}}
{{1,3},{2,3},{1,2,3}}
MATHEMATICA
Table[Length[Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Select[Tuples[#], UnsameQ@@#&]!={}&]], {n, 0, 3}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 05 2023
EXTENSIONS
a(6)-a(8) from Christian Sievers, Jul 28 2024
STATUS
approved