Probability and Algorithms
Caltech CS150, Winter 2006
Leonard J. Schulman
TA: ChihKai (Kevin) Ko.
Problem set 5
Out Wednesday February 22. Due Monday March 6.
Week 6 reading: MU 6.76.8, 7.17.4, proof of the PerronFrobenius theorem in notes of Andries
Brouwer on the class website.
This assignment is longer than usual so I’m allowing two weeks for it.
Please get started early
though. I haven’t assigned work on the PerronFrobenius theorem but it is an important theorem
with an interesting proof, and well worth learning. (You can also find the proof elsewhere, e.g., some
notes online by Mike Boyle.)
Exercises: MU 6.16, 6.18, 7.1, 7.6, 7.8, 7.13, 7.17, 7.19, 7.26.
Also the following two exercises:
(A) Let the
van der Waerden
function
W
(
k
)
, for nonnegative integer
k
, be the least nonnegative integer
n
so that if
{
1
, . . . , n
}
is twocolored, there exists a monochromatic arithmetic progression with
k
terms. Remarkably,
W
(
k
)
is finite for all
k
; the upper bound is immense and the true value is far
from known. In this exercise you are asked to show a lower bound on
W
(
k
)
. Try coloring
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 Algorithms, Lovász Local Lemma, Leonard J. Schulman, trajectory code

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