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

Abstract

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 ∈ 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 Turán numbers. It is easy to show that, for n ≥ k(p - 1)₁ (mathematical equation presented) For k = 4, we prove that equality holds if 61(n, 4, p, 2) < T(n, 2, p) + (mathematical equation presented) + 4. Moreover, we use this result to prove that L(n, 4, 3, 2) = 1(n, 4, 3, 2) if n ≥ 8.