On the structure of (v,k,t) trades (Q1076463)

Let V be a set of v treatments. Then \(v\Sigma\) t is the collection of all t-subsets of V. Let k,t be positive integers such that \(t<k<v\). The elements of \(v\Sigma\) k are called blocks. A (v,k,t) trade of volume s consists of two disjoint collections \(T_ 1\) and \(T_ 2\) each consisting of s blocks such that for every element of \(v\Sigma\) t the number of blocks containing this element is the same in both \(T_ 1\) and \(T_ 2\). If s equals zero the trade is said to be void. The set of treatments covered by \(T_ 1\) is the same as that for \(T_ 2\) and this common set is called the foundation of the trade. The author motivates the study of the structure of (v,k,t) trades by observing that when a t-(v,k,\(\lambda)\) design D contains the collection \(T_ 1\) of a (v,k,t) trade, then replacing the blocks of \(T_ 1\) in D by the blocks in \(T_ 2\) of the trade yields another t-(v,k,\(\lambda)\) design. The following results are proved: (i) The minimum foundation size of a nonvoid (v,k,t) trade is \(k+t+1.\) (ii) For \(v\geq k+t+1\), the minimum volume of a nonvoid (v,k,t) trade is \(2^ t.\) (iii) The nonvoid (v,k,t) trades with the minimum foundation size \(k+t+1\) and minimum volume \(2^ t\) exist and have a unique structure. (iv) For any v,k and t, \(t\geq 2\), there is no (v,k,t) trade of volume \(2^ t+1\).