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On Erdős’ theorem for monotonic subsequences. (English) Zbl 1129.05003

Variations of Erdős’ theorem about the longest monotonic subsequences of finite sequences of reals are discussed. The authors consider such subsequences with a specified first/last element from the sequence and relate their lengths to monotonic subsequences consisting of three consecutive elements. As an new application of Erdős’ theorem they give a combinatorial characterization of nonconvergent permutations where a permutation \(\pi\) of the set of positive integers \(\mathbb{N}\) is convergent, if for every convergent series \(\sum_{n\in \mathbb{N}}a_n\) of reals the series \(\sum_{n\in \mathbb{N}}a_{\pi(n)}\) is also convergent.

MSC:

05A05 Permutations, words, matrices
05D99 Extremal combinatorics
40A05 Convergence and divergence of series and sequences
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