Erdos-Ko-Rado Theorems of Labeled Sets

For k = (k ₁, ..., k _n ) ∈ N ⁿ, 1 ≤ k ₁ ≤... ≤ k _n , let L_k^r\mathcal{L}_k^r be the family of labeled r-sets on k given by L_k^r : = { { ( a₁ ,l_a1 ), ?,( a_r ,l_ar ) }:{ a₁ , ?a_r } í n ],l_ai ? k_ai ],i = 1, ?,r }\mathcal{L}_k^r : = \left\{ {\left\{ {\left( {a_1 ,l_{a_1 } } \right), \cdots ,\left( {a_r ,l_{a_r } } \right)} \right\}:\left\{ {a_1 , \cdots a_r } \right\} \subseteq \left n \right],l_{a_i } \in \left {k_{a_i } } \right],i = 1, \cdots ,r} \right\}. A family A\mathcal{A} of labeled r-sets is intersecting if any two sets in A intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.