\documentclass[12pt,ifthen]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{html}
\usepackage{hyperref}
\usepackage{tikz}
\newcommand{\OBS}{{\rm D}}
\newcommand{\D}{{\sf D}}
\newcommand{\N}{{\sf N}}
\newcommand{\Z}{{\sf Z}}
\newcommand{\Q}{{\sf Q}}
\newcommand{\R}{{\sf R}}
\newcommand{\reals}{{\sf R}}
\newcommand{\into}{{\rightarrow}}
\newcommand{\ceil}[1]{\lceil #1 \rceil}
\newcommand{\COL}{{\rm COL}}
\newcommand{\fractional}{{\rm frac}}

\usepackage{amsthm}
\newtheorem{theorem}{Theorem}

\begin{document}

\centerline{\bf CMSC 752 Homework 11} 
\centerline{\bf Morally Due Tue April 22, 2025} 
\centerline{\bf Dead Cat April 24}

\bigskip

\newif{\ifshowsoln}
\showsolntrue % comment out to NOT show solution inside \ifshowsoln and \fi blocks.

\smallskip

\begin{enumerate} 

\item 
(50 points) 
Show that every $\COL\colon [11]\times [11]\into [3]$ there is a mono rectangle. 

\newpage

\item
(50 points)
The shape below is called a {\it Soren}. 
In a Soren the following pairs-of-points have the same distance apart:

1 and 2

1 and 3

2 and 4

3 and 4

4 and 5.


Given a coloring, a {\it monochromatic Soren} is a Soren where all five vertices
have the same color. 

\bigskip

\begin{tikzpicture}
    \node[shape=circle,draw=black] (1) at (0,5) {1};
    \node[shape=circle,draw=black] (2) at (0,7) {2};
    \node[shape=circle,draw=black] (3) at (2,5) {3};
    \node[shape=circle,draw=black] (4) at (2,7) {4};
    \node[shape=circle,draw=black] (5) at (4,7) {5};

    \path [-, black,  thick](1) edge (2);
    \path [-, black,  thick](1) edge (3);
    \path [-, black,  thick](2) edge (4);
    \path [-, black,  thick](3) edge (4);
    \path [-, black,  thick](4) edge (5);

\end{tikzpicture}

Show that there exists a number $N$ such that, for all 2-colorings of $[N]\times [N]$
there exists a monochromatic Soren.
You may assume the following:

{\it For all $c$ there exists $S=SQ(c)$ such that for all $c$-colorings of $[S]\times[S]$
there exists a monochromatic square (all four corners the same color.)}

\end{enumerate}

\end{document}