The lotto numbers L(n, 4,p, 2)

摘要

An (n,k,p,t)-lotto design is an n-set N and a set B of k-subsets of N (called blocks) such that for each p-subset P of N, there is a block B 6 B for which |P ∩ B| > 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 and the numbers T(n,k,p) = L(n,k,p,k) are called Turan numbers. It is easy to show that, for n > k(p - 1), L(n,k,p,2) ≤1(n,k,p,2): = min a1+…a p-1 =n ai≥k = n(∑ from p-1 to i=1 C(ai, k, 2)) For k = 4, we prove that equality holds if 61(n,4,p, 2) < T(n, 2,p) + 「n/p-n」 + 4. Moreover, we use this result to prove that L(n,4,3,2) = l(n, 4,3,2) if n > 8.