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\centerline{\bf Homework 5, Morally Due Tue Mar 10, 2020, 3:30PM}

COURSE WEBSITE: \url{http://www.cs.umd.edu/~gasarch/858/S18.html}

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\begin{enumerate}

\item
(0 points)
What is your name?  Write it clearly.
Staple your HW. 
What type of midterm will there be?

\item (35 points) The \emph{Manhattan metric} is the following metric
  in $\R^{2}$: the distance between a point $p_{1} = (x_{1},y_{1})$
  and $p_{2} = (x_{2},y_{2})$ is
  $|p_{2}-p_{1}|_{1} = |x_{2}-x_{1}| + |y_{2}-y_{1}|$.
  (Note that this is the same as the the \emph{$L_{1}$ norm})
  
Prove the following using the 2-ary Can Ramsey Theorem:

If $X\subseteq \R^{2}$ is a countable set of points with the
\emph{Manhattan metric}, there exists a countable $Y\subseteq X$ such
that every pair of points in $Y$ has a different distance in the
Manhattan metric.

\item
(35 points)
(For this problem assume that there is NO cardinality between
countable and the cardinality of the reals.)
We say $|X|=|\R|$ to mean that $X$ and $\R$ are the same size,
so there is a bijection between them.

Prove the following using a Maximal Set argument:

If $X\subseteq \R^3$, $|X|=|\R|$,
no four on the same plane, there exists 
$Y\subseteq\R^3$, $|Y|=|\R|$,
such that 
every 4-subset of $Y$ yields a different volume.


\item (30 points) Prove that for all 2-colorings of the $5\times 5$
  grid, there is a monochromatic rectangle.

  Recall that the $5\times 5$ grid is
  \[
    \{ (a,b) : a,b\in \N \text{ and } 1\le a,b \le 5 \}
  \]
  and a \emph{monochromatic rectangle} is a set of four points that
  are the corners of a rectange, i.e.\ a set of points
  \[
    \{ (x,y), (x+c,y), (x,y+d), (x+c,y+d) \}
  \]
  (for some $x,y,c,d\in[5]$) that are all the same color.

\end{enumerate}

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