On Erdos' theorem for monotonic subsequences

• Autorzy: Wituła R. , Słota D. , Seweryn R.
• Języki publikacji: EN

Abstrakty

EN

In this paper finite one-one sequences of reals are studied. We consider the strengthening of famous Erdös'theorem. We discuss the lengths of the largest decreasing and increasing subsequences of the given sequences. Also, we study the length of the largest monotonic subsequences, which the first or the last elment is equal to a given elment ai of the sequence a. What is particulary important is the connection betwen estimation of these values with the problem of the existence of the3-elements monotonic subsequences of a having the form{ak, ak+1, ak+2}. Moreover, we introduce some conditons which are sufficient to the existence of such 3-elments subsequences of sequence a. As a new example of the application of Erdös' theorem for monotonic subsequences we give a combinatoric characterization of divergent permutations.

Słowa kluczowe

EN

Erdos theorem; permutations; divergent permutations; monotonic subsequences

PL

twierdzenie Erdosa; permutacje; permutacje rozbieżne; podciągi monotoniczne

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2007
• Tom: Vol. 40, nr 2
• Strony: 229--259
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Wituła R.

autor

Słota D.

autor

Seweryn R.

• Institute of Mathematics Silesian University of Technology Kaszubska 23 44-100 Gliwice, Poland,

Bibliografia

• [1] M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, New York, 2001.
• [2] P. Erdös, G. Szekeres, A combinatorial problem in geometry, Compositio Math. (1935), 463-470.
• [3] F. W. Levi, Rearrangement of convergent series, Duke Math. J. 13 (1946), 579-585.
• [4] B. S. Steczkin, On monotonie subseąuences in permutations of the set {l, 2,..., n}, Mat. Zametki 13 (1973), 511-514 (in Russian).
• [5] R. Wituła, The Riemann theorem and divergent permutations, Colloą. Math. 69 (1995), 275-287.
• [6] R. Wituła, Convergence-preserving functions, Nieuw Arch. Wisk. 13 (1995), 31-35.
• [7] R. Wituła, On the set of limit points of partial sums of series rearranged by a given divergent permutation, J. Math. Anal. Appl., to appear.
• [8] R. Wituła, M. J. Przybyła, The strongly and weakly divergent permutations, Demonstratio Math. 39 (2006), 107-116.
• [9] R. Wituła, D. Słota, The convergence classes o f divergent permutations, to appear.