Changed layout for more manageable volumes

1 files changed, 0 insertions, 508 deletions

diff --git a/2.3.4.3.lyx b/2.3.4.3.lyx
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-#LyX 2.4 created this file. For more info see https://www.lyx.org/
-\lyxformat 620
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-\begin_layout Standard
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-\begin_layout Plain Layout
-TODO 1,
- 3,
- 6 (2pp.,
- 0:46)
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-
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-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc1[M10]
-\end_layout
-
-\end_inset
-
-The text refers to a set 
-\begin_inset Formula $S$
-\end_inset
-
- containing finite sequences of positive integers,
- and states that this set is 
-\begin_inset Quotes eld
-\end_inset
-
-essentially an oriented tree.
-\begin_inset Quotes erd
-\end_inset
-
- What is the root of this oriented tree,
- and what are the arcs?
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-
-\begin_layout Standard
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-status open
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-\begin_layout Plain Layout
-
-
-\backslash
-answer 
-\end_layout
-
-\end_inset
-
-The vertices are the elements of 
-\begin_inset Formula $S$
-\end_inset
-
-,
- the root is 
-\begin_inset Formula $\emptyset$
-\end_inset
-
-,
- and the arcs go from 
-\begin_inset Formula $(x_{1},\dots,x_{n})$
-\end_inset
-
- to 
-\begin_inset Formula $(x_{1},\dots,x_{n-1})$
-\end_inset
-
-,
- for every 
-\begin_inset Formula $(x_{1},\dots,x_{n})\in S\setminus\{\emptyset\}$
-\end_inset
-
-.
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-
-\begin_layout Standard
-\begin_inset ERT
-status open
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-\begin_layout Plain Layout
-
-
-\backslash
-rexerc3[M23]
-\end_layout
-
-\end_inset
-
-If it is possible to tile the upper right quadrant of the plane when given an 
-\emph on
-infinite
-\emph default
- set of tetrad types,
- is it always possible to tile the whole plane?
-\end_layout
-
-\begin_layout Standard
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-status open
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-\begin_layout Plain Layout
-
-
-\backslash
-answer 
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-
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-
-No.
- For example,
- our tiles might be of the form
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-
-
-\backslash
-begin{center}
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-
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-\backslash
-begin{tikzpicture}
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-
-
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-draw (0,0) -- (0,2) -- (2,2) -- (2,0) -- (0,0) -- (2,2)
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-      (0,2) -- (2,0)
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-      (0.5,1) node{$n$} (1,0.5) node{$n$}
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-      (1.5,1) node{$n+1$} (1,1.5) node{$n+1$};
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-
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-
-for 
-\begin_inset Formula $n\in\mathbb{N}$
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-
-.
- Then in the 
-\begin_inset Formula $(x,y)$
-\end_inset
-
- square we might place the tile with 
-\begin_inset Formula $n=\max\{x,y\}$
-\end_inset
-
- and this would tile the upper right quadrant,
- but there's obviously no way to tile the whole plane.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-rexerc6[M23]
-\end_layout
-
-\end_inset
-
-(Otto Schreier.) In a famous paper,
- B.
- L.
- van der Waerden proved the following theorem:
-\end_layout
-
-\begin_layout Quote
-
-\shape slanted
-If 
-\begin_inset Formula $k$
-\end_inset
-
- and 
-\begin_inset Formula $m$
-\end_inset
-
- are positive integers,
- and if we have 
-\begin_inset Formula $k$
-\end_inset
-
- sets 
-\begin_inset Formula $S_{1},\dots,S_{k}$
-\end_inset
-
- of positive integers with every positive integer included in at least one of these sets,
- then at least of the sets 
-\begin_inset Formula $S_{j}$
-\end_inset
-
- contains an arithmetic progression of length 
-\begin_inset Formula $m$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-(The latter statement means there exist integers 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $\delta>0$
-\end_inset
-
- such that 
-\begin_inset Formula $a+\delta$
-\end_inset
-
-,
- 
-\begin_inset Formula $a+2\delta$
-\end_inset
-
-,
- ...,
- 
-\begin_inset Formula $a+m\delta$
-\end_inset
-
- are all in 
-\begin_inset Formula $S_{j}$
-\end_inset
-
-.) If possible,
- use this result and the infinity lemma to prove the following stronger statement:
-\end_layout
-
-\begin_layout Quote
-
-\shape slanted
-If 
-\begin_inset Formula $k$
-\end_inset
-
- and 
-\begin_inset Formula $m$
-\end_inset
-
- are positive integers,
- there is a number 
-\begin_inset Formula $N$
-\end_inset
-
- such that if we have 
-\begin_inset Formula $k$
-\end_inset
-
- sets 
-\begin_inset Formula $S_{1},\dots,S_{k}$
-\end_inset
-
- of integers with every integer between 1 and 
-\begin_inset Formula $N$
-\end_inset
-
- included in at least one of these sets,
- then at least one of the sets 
-\begin_inset Formula $S_{j}$
-\end_inset
-
- contains an arithmetic progression of length 
-\begin_inset Formula $m$
-\end_inset
-
-.
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-
-\begin_layout Standard
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-status open
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-\begin_layout Plain Layout
-
-
-\backslash
-answer 
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-
-\end_inset
-
-It is enough to prove it when the set that contains every integer between 1 and 
-\begin_inset Formula $N$
-\end_inset
-
- is 
-\begin_inset Formula $S_{1}$
-\end_inset
-
-.
- We define a tree whose vertices are the tuples 
-\begin_inset Formula $(S_{1},\dots,S_{k})$
-\end_inset
-
- of finite sets of integers such that,
- if 
-\begin_inset Formula $n=\max\bigcup_{i=1}^{k}S_{i}$
-\end_inset
-
-,
- every positive integer up to 
-\begin_inset Formula $n$
-\end_inset
-
- is in one of the sets,
- and such that no set contains an arithmetic progression of length 
-\begin_inset Formula $m$
-\end_inset
-
-.
- The edges go from one such tuple to 
-\begin_inset Formula $(S_{1}\setminus\{n\},\dots,S_{k}\setminus\{n\})$
-\end_inset
-
-,
- so the root is 
-\begin_inset Formula $(\emptyset,\dots,\emptyset)$
-\end_inset
-
- and 
-\begin_inset Formula $n$
-\end_inset
-
- is the height of the node in the tree.
-\end_layout
-
-\begin_layout Standard
-Since each vertex contains a finite number of children (
-\begin_inset Formula $2^{k}-1$
-\end_inset
-
-,
- corresponding to the ways of adding the next number to one or more of the sets),
- it follows that if this tree were infinite,
- there would be an infinite path.
- By van der Waerden theorem,
- the component-wise union of this path would give us a tuple 
-\begin_inset Formula $(S_{1},\dots,S_{k})$
-\end_inset
-
- such that some 
-\begin_inset Formula $S_{j}$
-\end_inset
-
- contains an arithmetic progression of length 
-\begin_inset Formula $m$
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-
-,
- so by construction one of the nodes in the path would follow this condition.
-\begin_inset Formula $\#$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Thus the tree is finite and we just need to take 
-\begin_inset Formula $N$
-\end_inset
-
- as one plus the depth of the tree.
-\end_layout
-
-\end_body
-\end_document