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On Erdos' theorem for monotonic subsequences

Wituła R., Słota D., Seweryn R.

Demonstratio Mathematica

2007

Vol. 40, nr 2

229-259

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.