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The Erds–Ko–Rado Theorem for Integer Sequences
The Erd˝os–Ko–Rado Theorem
for Integer Sequences
Peter Frankl
CNRS, ER 175 Combinatoire,
54 Bd Raspail, 75006 Paris, France
Norihide Tokushige
College of Education, Ryukyu University,
Nishihara, Okinawa, 903-0213, Japan
hide@edu.u-ryukyu.ac.jp
Abstract
For positive integers n, q, t we determine the maximum number of integer
sequences (a , . . . , a ) which satisfy 1 ≤ a ≤ q for 1 ≤ i ≤ n, and any two
1 n i
sequences agree in at least t positions. The result gives an affirmative answer
to a conjecture of Frankl and F¨uredi.
1 Introduction
Let n, q, t be positive integers with q ≥ 2, n ≥ t, and let [q] := {1, 2, . . . , q}. Then
H ⊂ [q]n is a set of integer sequences (a , . . . , a ), 1 ≤ a ≤ q . We say that H is
1 n i
t-intersecting if any two sequences intersects in at least t positions, more precisely,
|{i : a = a }| ≥ t holds for all (a , . . . , a ), (a , . . . , a ) ∈ H. In this paper, we
i i 1 n 1 n
determine the exact value of the following function.
f (n, q, t) := max{|H| : H ⊂ [q]n , H is t-intersecting}.
[n]
A family A ⊂ 2 is called t-intersecting if |A∩A | ≥ t holds for all A, A ∈ A. Define
a weighted size of A by w (A) := A∈A (q − 1)n−|A |. Using a shifting technique, it is
not difficult to check the following:
Lemma 1 (Proposition
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