# System of common representatives

From Encyclopedia of Mathematics

*system of simultaneous representatives*

A set of cardinality which is a system of different representatives for each of the families of subsets of a given set , each of which consists of elements. Suppose that , that is finite, let , , and let . A system of common representatives for and exists if and only if no sets of the family are contained in fewer than sets of , for each . This theorem is valid also for infinite , provided all the subsets in the families and are finite. Conditions for the existence of a system of common representatives are known for , but are more complicated to formulate.

#### References

[1] | M. Hall, "Combinatorial theory" , Wiley (1986) |

#### Comments

#### References

[a1] | H.J. Ryser, "Combinatorial mathematics" , Math. Assoc. Amer. (1963) |

[a2] | L. Mirsky, "Transversal theory" , Acad. Press (1971) |

[a3] | M. Aigner, "Combinatorial theory" , Springer (1979) pp. Chapt. II (Translated from German) |

**How to Cite This Entry:**

System of common representatives.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=System_of_common_representatives&oldid=16892

This article was adapted from an original article by V.E. Tarakanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article