Question

Please help with #6

'rove: Given a sequence of n2 +1 distinct integers, either there is an increasing subsequence of n+1 terms or a decreasing subsequence of n +1 terms.

Answer 1

Given a seq^n of n^2 + 1 distinct integers (a_1, a_2, ..., a_n^2 + 1) Associate an ordered pair (i_k, d_k) with each term of the seq^n where i_k is the length of the longest increasing subseq^n starting at a_k & d_k is the length of the longest decreasing subsequence starting at a_k. We will prove it by contradiction Now suppose there are no increasing or decreasing ssubsequences of length n + 1 or greater then i_k & d_k are oth positive integers lessthanorequalto n, for k = 1 to n^2 + 1 there are n^2 ordered pairs for (i_k, d_k) By the pigeon hole principle, since we have n^2 + 1 ordered pairs two must be identical. Exist as & a_t in seq^n st i_s = i_t & ds = dt '.' the terms are distinct in the sequence a_s < a_t or a_s > a_t If a_s < a_t, an increasing subseq^n of length i_t + 1