Information AboutProperty B |
| CATEGORIES ABOUT PROPERTY B | |
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VALUES OF ''M''(''N'') It is known that ''m''(1) = 1, ''m''(2) = 3, and ''m''(3) = 7 (as can by seen by the following examples); the value of ''m''(4) is not known, although an upper bound of 23 (Seymour, Toft) and a lower bound of 21 (Manning) have been proven. At the time of this writing (August 2004), 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'' = ; ''m''(2) : For ''n'' = 2, set ''X'' = {1, 2, 3} and ''C'' = . Then C does not have Property B, so ''m''(2) <= 3. However, ''C'' ; ''m''(3) : For ''n'' = 3, set ''X'' = {1, 2, 3, 4, 5, 6, 7}, and ''C'' = (the Steiner Triple System ''S''7); ''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'' REFERENCES
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