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''transversal design, transversal scheme, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t0939902.png" />-system''

A system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t0939903.png" /> of sets defined for a given collection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t0939904.png" /> pairwise-disjoint finite sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t0939905.png" />, each of which has cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t0939906.png" />. Namely: a transversal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t0939907.png" /> is a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t0939908.png" /> sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t0939909.png" /> (blocks or transversals), containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399010.png" /> elements each and such that:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399011.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399013.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399014.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399015.png" />.

In a transversal design, any two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399018.png" />, occur together in exactly one block. The existence of a transversal design <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399019.png" /> is equivalent to the existence of an [[Orthogonal array|orthogonal array]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399020.png" />.

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

A set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399021.png" /> transversals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399022.png" /> is called parallel if no two of them intersect. If a transversal design <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399023.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399024.png" /> (or more) parallel classes, then it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399025.png" />.

Some of the basic properties of transversal systems are:

a) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399027.png" /> exist, then so does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399028.png" />;

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399029.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399030.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399031.png" /> making up the design are called point classes or point groups.

The existence of a transversal design <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399032.png" /> is equivalent to the existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399033.png" /> mutually [[Orthogonal Latin squares|orthogonal Latin squares]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399034.png" />. If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399035.png" /> exists, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399036.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399037.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399038.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399039.png" /> with point classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399041.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399042.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399043.png" />-subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399044.png" /> and put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399045.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399046.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399047.png" /> for every block <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399049.png" />. Assume the existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399050.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399051.png" /> and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399052.png" /> for each block <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399054.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399055.png" /> also exists.

Transversals designs (and their dual structures, nets) are also of considerable geometric and algebraic interest. For instance, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399056.png" /> is equivalent to an affine or [[Projective plane|projective plane]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399057.png" />, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399058.png" /> is basically the same as a [[Quasi-group|quasi-group]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093990/t09399059.png" />. 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>