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''system of simultaneous representatives''

A set  $  R $
of cardinality  $  m $
which is a [[System of different representatives|system of different representatives]] for each of the  $  t $
families of subsets  $  {\mathcal F} _ {1} \dots {\mathcal F} _ {t} $
of a given set  $  S $, 
each of which consists of  $  m $
elements. Suppose that  $  t = 2 $, 
that  $  m $
is finite, let  $  {\mathcal F} _ {1} = \{ F _ {1} ^ { \prime } \dots F _ {m} ^ { \prime } \} $, 
$  {\mathcal F} _ {2} = \{ F _ {1} ^ { \prime\prime } \dots F _ {m} ^ { \prime\prime } \} $, 
and let  $  S = \cup _ {i = 1 }   ^ {m} F _ {i} ^ { \prime } = \cup _ {i = 1 }   ^ {m} F _ {i} ^ { \prime\prime } $. 
A system of common representatives for  $  {\mathcal F} _ {1} $
and  $  {\mathcal F} _ {2} $
exists if and only if no  $  k $
sets of the family  $  {\mathcal F} _ {1} $
are contained in fewer than  $  k $
sets of  $  {\mathcal F} _ {2} $, 
for each  $  k = 1 \dots m $. 
This theorem is valid also for infinite  $  m $, 
provided all the subsets in the families  $  {\mathcal F} _ {1} $
and  $  {\mathcal F} _ {2} $
are finite. Conditions for the existence of a system of common representatives are known for  $  t > 2 $, 
but are more complicated to formulate.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Hall,   "Combinatorial theory" , Wiley  (1986)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.J. Ryser,   "Combinatorial mathematics" , Math. Assoc. Amer.  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Mirsky,   "Transversal theory" , Acad. Press  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Aigner,   "Combinatorial theory" , Springer  (1979)  pp. Chapt. II  (Translated from German)</TD></TR></table>