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| {{for|the B-property in finite group theory|B-theorem}}
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| In [[mathematics]], '''Property B''' is a certain [[set theory|set theoretic]] property. Formally, given a [[finite set|finite]] set ''X'', a collection ''C'' of [[subset]]s of ''X'', all of size ''n'', has Property B if we can partition ''X'' into two disjoint subsets ''Y'' and ''Z'' such that every set in ''C'' meets both ''Y'' and ''Z''. The smallest number of sets in a collection of sets of size ''n'' such that ''C'' does not have Property B is denoted by ''m''(''n'').
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| The property gets its name from mathematician [[Felix Bernstein]], who first introduced the property in 1908.
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| == Values of ''m''(''n'') ==
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| 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.
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| ; ''m''(1)
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| : For ''n'' = 1, set ''X'' = {1}, and ''C'' = <nowiki>{{1}}</nowiki>. Then C does not have Property B.
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| ; ''m''(2)
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| : For ''n'' = 2, set ''X'' = {1, 2, 3} and ''C'' = {{1, 2}, {1, 3}, {2, 3}}. 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. | |
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| ; ''m''(3)
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| : 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. | |
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| ; ''m''(4)
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| : Seymour (1974) constructed a hypergraph on 11 vertices with 23 edges without Property B, which shows that ''m''(4) <= 23. Manning (1995) proved that ''m''(4) >= 20. | |
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| == Asymptotics of ''m''(''n'') ==
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| Erdős (1963) proved that for any collection of fewer than <math>2^{n-1}</math> sets of size ''n'', there exists a 2-coloring in which no set is monochromatic. The proof is simple: Consider a random coloring. The probability that an arbitrary set is monochromatic is <math>2^{-n+1}</math>. By a [[union bound]], the probability that there exist a monochromatic set is less than <math>2^{n-1}2^{-n+1} = 1</math>. Therefore, there exists a good coloring.
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| Erdős (1964) constructed an ''n''-uniform graph with <math>O(2^n \cdot n^2)</math> edges which does not have property B, establishing an upper bound. Schmidt (1963) proved that every collection of at most <math>n/(n+4)\cdot 2^n</math> has property B. Erdős and Lovász conjectured that <math>m(n) = \theta(2^n \cdot n)</math>. Beck in 1978 improved the lower bound to <math>m(n) = \Omega(n^{1/3}2^n)</math>. In 2000, Radhakrishnan and Srinivasan improved the lower bound to <math>m(n) = \Omega(2^n \cdot \sqrt{n / \log n})</math>. They used a clever probabilistic algorithm.
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| == References ==
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| *{{citation|last=Bernstein|first=F.|title=Zur theorie der trigonometrische Reihen|journal=Leipz. Ber.|volume=60|year=1908|pages=325–328}}.
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| *{{citation|last=Erdős|first=Paul|title=On a combinatorial problem|journal=Nordisk Mat. Tidskr.|year=1963|pages=5–10|volume=11}}
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| *{{cite doi|10.1007/BF01897152}}
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| *{{cite doi|10.1007/BF01897145}}
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| *{{citation|doi=10.1112/jlms/s2-8.4.681|last=Seymour|first=Paul|authorlink=Paul Seymour (mathematician)|title=A note on a combinatorial problem of Erdős and Hajnal|journal=Bulletin of the London Mathematical Society|volume=8|year=1974|pages=681–682}}.
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| *{{citation|last=Toft|first=Bjarne|contribution=On colour-critical hypergraphs|title=Infinite and Finite Sets: To Paul Erdös on His 60th Birthday|editor1-first=A.|editor1-last=Hajnal|editor1-link=András Hajnal|editor2-first=Richard|editor2-last=Rado|editor2-link=Richard Rado|editor3-first=Vera T.|editor3-last=Sós|publisher=North Holland Publishing Co.|year=1975|pages=1445–1457}}.
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| *{{citation|doi=10.1090/S1079-6762-95-03004-6|first=G. M.|last=Manning|title=Some results on the ''m''(4) problem of Erdős and Hajnal|journal=[[Electronic Research Announcements of the American Mathematical Society]]|volume=1|year=1995|pages=112–113}}.
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| *{{citation|first=J.|last=Beck|title=On 3-chromatic hypergraphs|journal=Discrete Mathematics|volume=24|issue=2|pages=127–137|year=1978|doi=10.1016/0012-365X(78)90191-7}}.
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| *{{citation|doi=10.1002/(SICI)1098-2418(200001)16:1<4::AID-RSA2>3.0.CO;2-2|first1=J.|last1=Radhakrishnan|first2=A.|last2=Srinivasan|title=Improved bounds and algorithms for hypergraph 2-coloring|url=http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=00743519|journal=Random Structures and Algorithms|volume=16|issue=1|pages=4–32|year=2000}}.
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| *{{citation|first=E. W.|last=Miller|title=On a property of families of sets|journal=Comp. Rend. Varsovie|year=1937|pages=31–38}}.
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| *{{citation|doi=10.1007/BF02066676|first1=P.|last1=Erdős|author1-link=Paul Erdős|first2=A.|last2=Hajnal|author2-link=András Hajnal|title=On a property of families of sets|journal=Acta Math. Acad. Sci. Hung.|volume=12|year=1961|pages=87–123}}.
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| *{{citation|doi=10.4153/CMB-1969-107-x|first1=H. L.|last1=Abbott|first2=D.|last2=Hanson|title=On a combinatorial problem of Erdös|journal=[[Canadian Mathematical Bulletin]]|volume=12|year=1969|pages=823–829}}
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| .
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| [[Category:Set families]]
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