Editing Steiner system (section)

== Properties ==
It is clear [[logical consequence|from the definition]] of {{math|S(''t'', ''k'', ''n'')}} that <math>1 < t < k < n</math>. (Equalities, while technically possible, lead to trivial systems.)

If {{math|S(''t'', ''k'', ''n'')}} exists, then taking all blocks containing a specific element and discarding that element gives a ''derived system'' {{math|S(''t''&minus;1, ''k''&minus;1, ''n''&minus;1)}}. Therefore, the existence of {{math|S(''t''&minus;1, ''k''&minus;1, ''n''&minus;1)}} is a necessary condition for the existence of {{math|S(''t'', ''k'', ''n'')}}.

The number of {{math|''t''}}-element subsets in {{math|S}} is <math>\tbinom n t</math>, while the number of {{math|''t''}}-element subsets in each block is <math>\tbinom k t</math>. Since every {{math|''t''}}-element subset is contained in exactly one block, we have <math>\tbinom n t = b\tbinom k t</math>, or 
:<math>b = \frac{\tbinom nt}{\tbinom kt} = \frac{n(n-1)\cdots(n-t+1)}{k(k-1)\cdots(k-t+1)},</math>
where {{math|''b''}} is the number of blocks. Similar reasoning about {{math|''t''}}-element subsets containing a particular element gives us <math>\tbinom{n-1}{t-1}=r\tbinom{k-1}{t-1}</math>, or 
:<math>r=\frac{\tbinom{n-1}{t-1}}{\tbinom{k-1}{t-1}}</math> =<math>\frac{(n-t+1)\cdots(n-2)(n-1)}{(k-t+1)\cdots(k-2)(k-1)},</math>
where {{math|''r''}} is the number of blocks containing any given element. From these definitions follows the equation <math>bk=rn</math>. It is a necessary condition for the existence of {{math|S(''t'', ''k'', ''n'')}} that {{math|''b''}} and {{math|''r''}} are integers. As with any block design, [[Fisher's inequality]] <math>b\ge n</math> is true in Steiner systems.

Given the parameters of a Steiner system {{math|S(''t, k, n'')}} and a subset of size <math>t' \leq t</math>, contained in at least one block, one can compute the number of blocks intersecting that subset in a fixed number of elements by constructing a [[Pascal triangle]].{{sfn|Assmus|Key|1992|page=8}}  In particular, the number of blocks intersecting a fixed block in any number of elements is independent of the chosen block.

The number of blocks that contain any ''i''-element set of points is:
:<math> \lambda_i =  \left.\binom{n-i}{t-i} \right/ \binom{k-i}{t-i} \text{ for } i = 0,1,\ldots,t, </math>

It can be shown that if there is a Steiner system {{math|S(2, ''k'', ''n'')}}, where {{math|''k''}} is a [[prime power]] greater than 1, then {{math|''n''}} <math>\equiv</math> 1 or {{math|''k'' (mod ''k''(''k''&minus;1))}}. In particular, a Steiner triple system {{math|S(2, 3, ''n'')}} must have {{math|1=''n'' = 6''m'' + 1 or 6''m'' + 3}}. And as we have already mentioned, this is the only restriction on Steiner triple systems, that is, for each [[natural number]] {{math|''m''}}, systems {{math|S(2, 3, 6''m'' + 1)}} and {{math|S(2, 3, 6''m'' + 3)}} exist.