An Asymptotic Complete Intersection Theorem for Chain Products

LetNℓ(n,k) be the set of alln-tuples over the alphabet {0, 1, . . . ,k} whose component sum equalsℓ. A subsetF ⊆ Nℓ(n,k) is called at-intersecting family if every two tuples inFhave nonzero entries in at leasttcommon coordinates. We determine the maximum size of at-intersecting family inN⌊λn⌋(n,k) asymptotically for all fixedλ( 0 < λ < k) andn → ∞.