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MA341C — Seminar on Proofs from THE BOOK Trinity College Dublin

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Assignment 1

MA341C — Seminar on Proofs from THE BOOK Trinity College Dublin

NAME AND SURNAME: . . . . STUDENT NUMBER: . . . NUMBER OF PAGES: . . . .

Note:solutions to this assignment are due by 11am on Wednesday, September 19th. Please attach a cover sheet with a declaration (http://tcd-ie.libguides.com/plagiarism/declaration) confirming that you know and understand College rules on plagiarism. All exercises are weighed equally unless otherwise stated.

Exercise 1.Show, by providing an example, that for any choice of natural num- bersm,n Ê2 there is a sequence ofmn real numbers with no decreasing sub- sequence of lengthm+1 nor any increasing subsequence of lengthn+1.

Exercise 2.Prove that if we draw a 3×7 grid as in the figure below, with vertices coloured purple and blue, there is one monochromatic rectangle with its edges parallel to the grid.

Exercise 3.Fix a natural numbern and letσbe a permutation onn letters. We sayi ∈[n] is a fixed point ofσifσ(i)=i, and write fix(σ) for the number of such fixed points. Use double counting to show that

1 n!

X

σ∈Sn

fix(σ)=1.

That is, the expected number of fixed points of a permutation is 1.

Exercise 4.Suppose thattis a real number andNis a positive integer. Show that among theN numbers t, 2t, . . . ,N t there is at least one which differs by at most 1/Nfrom an integer.

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