On erdös' theorem for monotonic subsequences

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