Editing Property B (section)

{{for|the B-property in finite group theory|B-theorem}}
[[File:Property b.svg|thumb|A [[Hypergraph#Hypergraph coloring|2-coloring]] of a hypergraph, equivalent to a collection C with Property B.]]
In [[mathematics]], '''Property B''' is a certain [[set theory|set theoretic]] property. Formally, given a [[finite set]] ''X'', a collection ''C'' of [[subset]]s of ''X'' 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 property gets its name from mathematician [[Felix Bernstein (mathematician)|Felix Bernstein]], who first introduced the property in 1908.<ref>{{citation|last=Bernstein|first=F.|title=Zur theorie der trigonometrische Reihen|journal=Leipz. Ber.|volume=60|year=1908|pages=325–328}}.</ref>

Property B is equivalent to [[Hypergraph#Hypergraph coloring|2-coloring]] the [[hypergraph]] described by the collection ''C''. A hypergraph with property B is also called '''2-colorable'''.<ref name="lp">{{Cite Lovasz Plummer}}</ref>{{rp|468}} Sometimes it is also called '''bipartite''', by analogy to the [[bipartite graph]]s.
Property B is often studied for uniform hypergraphs (set systems in which all subsets of the system have the same cardinality) but it has also been considered in the non-uniform case.<ref>{{citation
 | last = Beck | first = J. | authorlink = József Beck
 | doi = 10.1016/0012-365X(78)90191-7
 | issue = 2
 | journal = Discrete Mathematics
 | mr = 522920
 | pages = 127–137
 | title = On 3-chromatic hypergraphs
 | volume = 24
 | year = 1978| doi-access = free
 }}</ref>

The problem of checking whether a collection ''C'' has Property B is called the [[set splitting problem]].