Wituła, Roman; Słota, Damian; Seweryn, Ryszard On Erdős’ theorem for monotonic subsequences. (English) Zbl 1129.05003 Demonstr. Math. 40, No. 2, 239-259 (2007). 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. Reviewer: Dieter Rautenbach (Ilmenau) Cited in 3 Documents MSC: 05A05 Permutations, words, matrices 05D99 Extremal combinatorics 40A05 Convergence and divergence of series and sequences Keywords:monotonic subsequence; divergent permutations PDFBibTeX XMLCite \textit{R. Wituła} et al., Demonstr. Math. 40, No. 2, 239--259 (2007; Zbl 1129.05003)