Cross-intersecting families of labeled sets

For two positive integers n and p, let L-p be the family of labeled n-sets given by L-p - {{(1,l(1)), (2,l(2)), . . . , (n, l(n))} : l(i) is an element of [p], i - 1,2 . . . , n}. Families A and B are said to be cross-intersecting if A boolean AND B not equal empty set for all A is an element of A and B is an element of B. In this paper, we will prove that for p >= 4, if A and B are cross-intersecting sub families of L-p, then vertical bar A vertical bar vertical bar B vertical bar <= p(2n-2), and equality holds if and only if A and B are an identical largest intersecting subfamily of L-p.