The lotto numbers L ( n ,3, p ,2)

An ( n,k,p,t )‐lotto design is an n ‐set N and a set ${\cal B}$ of k ‐subsets of N (called blocks) such that for each p ‐subset P of N , there is a block $B \in {\cal B}$ for which $\left | P \cap B \right | \geq t$ . The lotto number L(n,k,p,t) is the smallest number of blocks in an ( n,k,p,t )‐lotto design. The numbers C(n,k,t) =  L(n,k,t,t) are called covering numbers. It is easy to show that, for n  ≥  k ( p  − 1),$$L(n,k,p,2)\leq \mathop {a_1 + \cdots + a_{p - 1} = n}\limits_{a_i k}^{\min } \left( {\sum\limits_{i = 1}^{p - 1} {C(a_i ,k,2)} } \right)$$ For k = 3, we prove that equality holds if one of the following holds: (i) n is large, in particular $ n\geq \big \{ {\matrix{(p - 1)(2p - 3)\quad {\rm if }n \not \equiv p(\bmod \;2),\cr (p - 1)(4p - 8)\quad {\rm if }n \equiv p(\bmod \;2),}}$(ii) $n \equiv p - 4,p - 3, \ldots ,3p - 1(\bmod 6(p - 1)),$(iii) 2 ≤ p ≤ 6. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 333–350, 2006