\documentclass[12pt,ifthen]{article}
\usepackage{amsmath}
\usepackage{amssymb}
%\usepackage{html}
\usepackage{hyperref}
\newcommand{\degg}{{\rm deg}}
\newcommand{\COL}{{\rm COL}}
\renewcommand{\S}{{\sf S}}
\newcommand{\N}{{\sf N}}
\newcommand{\Z}{{\sf Z}}
\newcommand{\R}{{\sf R}}
\newcommand{\into}{{\rightarrow}}
\newcommand{\PF}{{P^{finite}}}
\newcommand{\ceil}[1]{\lceil #1 \rceil}

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

\begin{document}

\centerline{\bf Homework 05} 

\centerline{Morally Due Tue March 1 at 3:30PM. Dead Cat March 3 at 3:30PM}

\begin{enumerate}

\item
(0 points)
What is your name?  Write it clearly.
When is the take-home midterm due?

\item
(30 points)
Prove the following (the finite 3-hypergraph Ramsey Theorem)
by using the infinite 3-hypergraph Ramsey Theorem. 

{\it For all $k,c$ there exists $n$ such that for all $\COL\colon\binom{[n]}{3}\into[c]$
there exists a homog set of size $k$. }

\vfill

\centerline{\bf GO TO NEXT PAGE}

\newpage

\item
(30 points)
We will look at the following statement which we call FCR (Finite Can Ramsey)

{\it For all $k,c$ there exists $n$ such that for all $\COL\colon\binom{[n]}{2}\into\omega$
either there is a homog set of size $k$ OR a min-homog set of size $k$ OR a max-homog set of size $k$
OR a rainbow set of size $k$.}

\begin{enumerate}
\item
(10 points) TRY to prove FCR from the infinite Ramsey Theorem. YOU WILL FAIL.
Where does it fail?
\item
(20 points) Prove FCR somehow (Hint: DO NOT try to finitize Mileti's proof. 
That can be done but is messy.) 
\end{enumerate}

\vfill

\centerline{\bf GO TO NEXT PAGE}

\newpage

\item
(40 points)
In this problem we will look at the following problems:

{\it Given $X\subseteq \R^1$ (or $\S^1$ or $\R^2$) of size $n$ show
there is a large subset of size $\Omega(f(n))$ (you will figure out the $f$)
where all of the distances are different. ($S^1$ is any circle.) }

 We will NOT use Can Ramsey.

Let $X$ be a set of points (could be in $\R^1$ or $S^1$ or $\R^2$).
Let $M\subseteq X$. $M$ is {\it d-maximal} if (1) every pair of distances is different and
(2) for all $p\in X-M$, statement (1) is false for $M\cup \{p\}$. 
(Note that we DO NOT have the color-degree bound we had in the past.) 
\begin{enumerate}
\item
(10 points) 
Find an increasing function $f$ such that the following is true:

{\it If $X\subseteq \R^1$ is a set of $n$ points then every d-maximal set is of size
$\Omega(f(n))$.}

\item 
(15 points) 
Find an increasing function $g$ such that the following is true:

{\it If $X\subseteq \S^1$ is a set of $n$ points then every d-maximal set is of size
$\Omega(g(n))$.}

\item 
(15 points) 
Find an increasing function $h$ such that the following is true:

{\it If $X\subseteq \R^2$ is a set of $n$ points then every d-maximal set is of size
$\Omega(h(n))$.}
\end{enumerate}
\end{enumerate}

\end{document}