SOME GREEDY t-INTERSECTING FAMILIES OF FINITE SEQUENCES

摘要

Let n, s_1, s_2,··· and s_n be positive integers. Assume M(s_1,s_2,··· ,s_n) = {(x_1,x_2,···,x_n) | 0 ≤ x_i ≤ s_i, x_i is an integer for each i}. For a = (a_1,a_2,··· ,a_n)∈M(s_1,s_2,··· ,s_n), F is contained in M(s_1,s_2,··· ,s_n), and A is contained in {1,2,··· ,n}, denote s_p(a) = {j | 1 ≤ j ≤ n, a_j ≥ p}, S_p(F) = {s_p(a) | a∈F}, and W_p(A) = p~(n-|A|)Π_(i∈A)(s_i-p). F is called an I_t~p-intersecting family if, for any a,b∈F, a_i∧b_i = min(a_i,b_i) ≥ p for at least t i's. F is called a greedy I_t~p-intersecting family if F is an I_t~p-intersecting family and W_p(A) ≥ W_p(B+A~c) for any A∈S_p(F) and any B is contained in A with |B| = t-1. In this paper, we obtain a sharp upper bound of |F| for greedy I_t~p-intersecting families in M(s_1,s_2,··· ,s_n) for the case 2p ≤ s_i (1 ≤ i ≤ n) and s_1 > s_2 > ··· > s_n.