On a Combinatorial Problem in Number Theory

نویسنده

  • L. G. CHOI
چکیده

1. Introduction One of the problems mentioned injl] (on p. 187) concerns the estimation of the magnitude of the largest number g(N) such that, from every sequence of N distinct natural numbers, one can always select a subsequence of g{N) integers with the property that no sum of two distinct integers of this subsequence belongs to the original sequence. It should be mentioned that the function 0 and N ^ N x (e). A very simple construction (see proof of Theorem 1 in §2) already shows that g(N) <^ iV* (SO that Theorem 1 alone suffices to establish Erdos' conjecture). The refinement (2) is obtained as a consequence of Theorem 3 below and the underlying idea in the construction in Theorem 1. However, we have included Theorem 2 as well, though the corresponding application of this theorem and the construction in Theorem 1 yields an estimate inferior to (2). This is because we think that Theorem 2 is, like Theorem 3, probably of independent interest. Concerning the function f(n) in Theorem 2, one may conjecture that/(w) = O(n

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تاریخ انتشار 1971