The Erdös-Ko-Rado Theorem for Integer Sequences

Abstract

For positive integers $n,k,t$ we investigate the problem how many integer sequences $( a_1 ,a_2 , \cdots ,a_n )$ we can take, such that $1\leqq a_i \leqq k$ for $1\leqq i\leqq n$, and any two sequences agree in at least t positions. This problem was solved by Kleitman (J. Combin. Theory, 1 (1966), pp. 209–214) for $k = 2$, and by Berge (in Hypergraph Seminar, Columbus, Ohio (1972), Springer-Verlag, New York, 1974) for $t = 1$. We prove that for $t\geqq 15$ the maximum number of such sequences is $k^{n - t} $ if and only if $k\geqq t + 1$.