LetV(n, k) denote the set of vectors of lengthn whose components are integersj with 1≦jk. For every two vectorsx, y inV(n, k), leta(x, y) stand for the number of subscriptsi withx i =y i . We prove that for every positive ε there is ann(ε) with the following property: ifn>n(ε) andk<n 1−ε then there is a setQ of at most (6+ε)(n logk)/(logn−logk) vectors inV(n, k) such that for every two distinct vectorsx, y inV(n, k) someq inQ hasa(q, x) ≠a(q, y).

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Dedicated to Paul Erdős on his seventieth birthday

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Chvátal, V. Mastermind. Combinatorica 3, 325–329 (1983).

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AMS subject classification (1980)

  • 05 B 99
  • 94 A 50, 90 D 99