Editing Property B (section)

=== Known values of m(n) ===
It is known that ''m''(1) = 1, ''m''(2) = 3, ''m''(3) = 7 (as can by seen by the following examples), and ''m''(4) = 23 (Östergård), although finding this result was the result of an exhaustive search. An upper bound of 23 (Seymour, Toft) and a lower bound of 21 (Manning) have been proven. At the time of this writing (March 2017), there is no [[OEIS]] entry for the sequence ''m''(''n'') yet, due to the lack of terms known.

; ''m''(1)
: For ''n'' = 1, set ''X'' = {1}, and ''C'' = <nowiki>{{1}}</nowiki>. Then C does not have Property B.

; ''m''(2) 
: For ''n'' = 2, set ''X'' = {1, 2, 3} and ''C'' = <nowiki>{{1, 2}, {1, 3}, {2, 3}}</nowiki> (a triangle). Then C does not have Property B, so ''m''(2) <= 3. However, ''C''<nowiki>'</nowiki> = {{1, 2}, {1, 3}} does (set ''Y'' = {1} and ''Z'' = {2, 3}), so ''m''(2) >= 3.

; ''m''(3)
: For ''n'' = 3, set ''X'' = {1, 2, 3, 4, 5, 6, 7}, and ''C'' = {{1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 7}, {5, 6, 1}, {6, 7, 2}, {7, 1, 3}} (the [[Steiner triple system]] ''S''<sub>7</sub>); ''C'' does not have Property B (so ''m''(3) <= 7), but if any element of ''C'' is omitted, then that element can be taken as ''Y'', and the set of remaining elements ''C''<nowiki>'</nowiki> will have Property B (so for this particular case, ''m''(3) >= 7).  One may check all other collections of 6 3-sets to see that all have Property B.

; ''m''(4)
: Östergård (2014) through an exhaustive search found ''m''(4) = 23. Seymour (1974) constructed a hypergraph on 11 vertices with 23 edges without Property B, which shows that ''m''(4) <= 23. Manning (1995) narrowed the floor such that ''m''(4) >= 21.