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''transversal design, transversal scheme,  $  T $-system''

A system  $  T _ {0} ( m, t) $
of sets defined for a given collection of  $  m $
pairwise-disjoint finite sets  $  S _ {1}, \dots, S _ {m} $, 
each of which has cardinality  $  t $. 
Namely: a transversal system  $  T _ {0} ( m, t) $
is a system of  $  t  ^ {2} $
sets  $  Y _ {1}, \dots, Y _ {t  ^ {2}  } $ (blocks or transversals), containing  $  m $
elements each and such that:

1)  $  | Y _ {j} \cap S _ {i} | = 1 $; 
$  i = 1, \dots, m $; 
$  j = 1, \dots, t  ^ {2} $;

2)  $  | Y _ {j} \cap Y _ {k} | \leq  1 $
for  $  j \neq k $.

In a transversal design, any two elements  $  a \in S _ {i} $
and  $  b \in S _ {j} $, 
$  i \neq j $, 
occur together in exactly one block. The existence of a transversal design  $  T _ {0} ( m, t) $
is equivalent to the existence of an [[Orthogonal array|orthogonal array]]  $   \mathop{\rm OA} ( t, m) $.

Transversal designs are used in recursive constructions of block designs (cf. [[Block design|Block design]]).

A set of  $  t $
transversals in  $  T _ {0} ( m, t) $
is called parallel if no two of them intersect. If a transversal design  $  T _ {0} ( m, t) $
contains  $  e $ (or more) parallel classes, then it is denoted by  $  T _ {e} ( m, t) $.

Some of the basic properties of transversal systems are:

a) if  $  T _ {d} ( m, s) $
and  $  T _ {e} ( m, t) $
exist, then so does  $  T _ {de} ( m, st) $;

b)  $  T _ {t} ( m - 1, t) $
exists if and only if  $  T _ {0} ( m, t) $
exists.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Hall,   "Combinatorial theory" , Wiley  (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Hanani,   "The existence and construction of balanced incomplete block designs"  ''Ann. Math. Stat.'' , '''32'''  (1961)  pp. 361–386</TD></TR></table>

====Comments====
The finite sets  $  S _ {1}, \dots, S _ {m} $
making up the design are called point classes or point groups.

The existence of a transversal design  $  T _ {0} ( m, t) $
is equivalent to the existence of  $  m - 2 $
mutually [[Orthogonal Latin squares|orthogonal Latin squares]] of order  $  t $. 
If a  $  T _ {0} ( m, t) $
exists, and  $  t > 1 $, 
then  $  m \leq  t + 1 $.

See also [[#References|[a1]]]–[[#References|[a3]]] for the recursive construction of and existence results for transversal designs (and of their generalization  "transversal design with holes" , [[#References|[a2]]]).

One of the most important results on recursive construction of transversal designs (due to R.M. Wilson [[#References|[a4]]], see also [[#References|[a1]]]) is as follows:

Let  $  D $
be a  $  T _ {0} ( k+ m, s) $
with point classes  $  P _ {1}, \dots, P _ {k} $
and  $  Q _ {1}, \dots, Q _ {m} $, 
let  $  T $
be a  $  t $-subset of  $  Q _ {1} \cup {} \dots \cup Q _ {m} $
and put  $  t _ {i} = | T \cap Q _ {i} | $
for  $  i = 1, \dots, m $, 
and  $  u _ {B} = | B \cap T | $
for every block  $  B $
of  $  D $. 
Assume the existence of  $  T _ {0} ( k , t _ {i} ) $
for  $  i = 1, \dots, m $
and of  $  T _ {u _ {B}  } ( k , n + u _ {B} ) $
for each block  $  B $
of  $  D $. 
Then  $  T _ {0} ( k, ns+ t ) $
also exists.

Transversals designs (and their dual structures, nets) are also of considerable geometric and algebraic interest. For instance, a  $  T _ {0} ( s+ 1, s) $
is equivalent to an affine or [[Projective plane|projective plane]] of order  $  s $, 
and a  $  T _ {0} ( 3, s) $
is basically the same as a [[Quasi-group|quasi-group]] of order  $  s $. 
Thus, the geometric and algebraic properties of transversal designs (including the study of their automorphism groups) have found considerable interest, cf. [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Beth,   D. Jungnickel,   H. Lenz,   "Design theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.E. Brouwer,   "Recursive constructions of mutually orthogonal Latin squares"  ''Ann. Discr. Math.'' , '''46'''  (1991)  pp. 149–168</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Jungnickel,   "Latin squares, their geometries and their groups: a survey"  D. Ray-Chaudhuri (ed.) , ''Coding Theory and Design Theory'' , ''IMA Vol. Math. Appl.'' , '''21''' , Springer  (1990)  pp. 166–225</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.M. Wilson,   "Concerning the number of mutually orthogonal Latin squares"  ''Discr. Math.'' , '''9'''  (1974)  pp. 181–198</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Hanani,   "Balanced incomplete block designs and related designs"  ''Discrete Math.'' , '''11'''  (1975)  pp. 255–369</TD></TR></table>