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

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

\begin{document}

\centerline{\bf CMSC 752 Homework 7} 
\centerline{\bf Morally Due Tue March 25, 2025} 
\centerline{\bf Dead Cat March 27}

\bigskip

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

\begin{enumerate}
\item
(50 points) 
In this problem we guide you through another proof of the Happy ending theorem. 

{\it For all $k\ge 3$ there exists an $n$ such that for set $X$ of $n$ points
in the plane there exists $Y\subseteq X$ with $|Y|=k$ and the points in $Y$ are
the vertices of a convex hull of size $k$.} 

We show that $n=R_3(k)$ suffices (for all $\COL\colon \binom{[n]}{3}\into [2]$
there is a homog set of size $k$). 

Let $X= \{p_1,\ldots,p_n\}$. 

Let $\COL\colon \binom{[n]}{3}$ be defined as follows:

$\COL(i<j<k)=$
\begin{itemize}
\item
RED if $p_i$-$p_j$-$p_k$ is CLOCKWISE
\item
BLUE if $p_i$-$p_j$-$p_k$ is COUNTER-CLOCKWISE
\end{itemize}

(Do you young folk even know what CLOCKWISE means, with your fancy
digital watches?) 

FINISH THE PROOF.

\newpage 

\item
(50 points) 
READ the statement of the 3-ary Can Ramsey theorem 

\url{https://www.cs.umd.edu/~gasarch/COURSES/752/S25/slides/aarycanramsey.pdf}

USE the 3-ary Can Ramsey Theorem to prove the following

{\it Let $X$ be a countably infinite set of points in $\R^2$ with no three points colinear.
Show that there is an infinite $Y\subseteq X$ such that every 3-set of $Y$ 
encloses a different area.} 

\newpage

\item
(0 points- Extra Credit)
(In this problem all colorings are RED-BLUE colorings.) 

Proof of Disprove: 
Show that for all $\COL \colon \binom{\omega^2}{2} \into [2]$ 
there is EITHER a RED homog $H\equiv \omega^2$ OR a BLUE homog set of size 1,000,000. 


\newpage

\item 
(0 points, DO THIS- there will be a quiz on this on Thursday.
If you are NOT in class that day I will email you the quiz later.) 
READ the Soren Brown-William Gasarch paper on application of Ramsey Theory
to history, and also the relevant parts of the papers and slides posted 
next to it. 

\end{enumerate}

\end{document}