\documentclass[11pt]{article}
\usepackage{fullpage}
\usepackage{url}
\usepackage{amsmath}

% This is a file of macros and definitions that may come up in
% ANY LATEX paper.  Typically I'll use this file and some other file in the
% directory of the paper, this one for general math things, that one for
% things specific to that paper.
%
% font's used and general paper things.
\font\tenrm=cmr10
\font\ninerm=cmr9
\font\eightrm=cmr8
\font\sevenrm=cmr7
% \font\title=cmbx10 scaled \magstep1 % extra big title font
\font\ss=cmss10 % used by \proof
\font\smallcaps=cmcsc10 % used to label Theorems, etc.
% imhibit black bars on overflows
%
\overfullrule=0pt
%
% today's date
%
%
% English words that I always italizice in papers.
% Some words that appear in math mode alot that I wasn roman
%

\newcommand{\AREA}{{\rm AREA}}
\newcommand{\ALG}{{\rm ALG}}
\newcommand{\ACK}{{\rm ACK}}
\newcommand{\VC}{{\rm VC}}
\newcommand{\AC}{{\rm AC}}
\newcommand{\ZL}{{\rm ZL}}
\newcommand{\WOP}{{\rm WOP}}
\newcommand{\tree}{{\rm tree}}
\newcommand{\SUBSEQ}{{\rm SUBSEQ}}
\newcommand{\minor}{{\preceq_m}}
\newcommand{\minorp}{{\preceq_m'}}
\newcommand{\cminor}{{\preceq_{\rm c\hbox{-}m}}}
\newcommand{\GM}{{\rm  GM}}
\newcommand{\TREE}{{\rm  TREE}}
\newcommand{\R}{\rm  R} 
\newcommand{\B}{\rm  B} 
\newcommand{\G}{\rm  G} 
\newcommand{\Y}{\rm  P} 

\newcommand{\IO}{\exists^\infty}
\newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}

\newcommand{\into}{{\rightarrow}}
\newcommand{\D}{{\sf D}}
\newcommand{\N}{{\sf N}}
\newcommand{\Z}{{\sf Z}}
\newcommand{\Q}{{\sf Q}}
\newcommand{\reals}{{\sf R}}
\newcommand{\qd}{{\rm qd}}
\newcommand{\GE}[1]{{\equiv^{G}_{#1}}}
\newcommand{\TE}[1]{{\equiv^{T}_{#1}}}
\newcommand{\TES}[1]{{\equiv^{T}_{#1,{\rm S}}}}
\newcommand{\TEB}[1]{{\equiv^{T}_{#1,{\rm BOOL(S)}}}}
\newcommand{\LL}{{\cal{L}}}
\newcommand{\onepe}{1+\epsilon}
\newcommand{\oneme}{1-\epsilon}
\newcommand{\cool}{cool}
\newcommand{\nze}{numzeros}
\newcommand{\non}{numones}
\newcommand{\LR}[2]{ {\rm LR}_{{#2}}({#1})}
\newcommand{\LH}[2]{ {\rm LH}({#1};{#2})}
\newcommand{\bbb}{{\gamma}}
\newcommand{\COL}{{\rm COL}}
\newcommand{\card}[1]{\#(#1)}
\newcommand{\st}{\mathrel{:}}
\newcommand{\fhat}{{\hat{f}}}
\newcommand{\Ahat}{{\hat{A}}}

\begin{document}

\centerline{\bf Homework 09, Morally Due 12:30PM, Tue Apr 14  2026}


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

\begin{enumerate}

\item
(0 points) What is your name? 


\vfill

\centerline{\bf GO TO NEXT PAGE}

\newpage



\item
(30 points) 
{\bf Recall} The infinite 3-ary Can Ramsey Theorem: For all $\COL\binom{\N}{3}\into [\omega]$
there is an infinite $H\subseteq \N$ such that one of the following 8 cases occurs. 

(If we write $\COL(x,y,z)$ then $x<y<z$.)

\begin{itemize}
\item
A $\emptyset$-homog set, so all triples are the same color.
\item
A $\{1\}$-homog set, so $\COL(x_1,x_2,x_3)=\COL(y_1,y_2,y_3)$ iff $x_1=y_1$. 
\item
A $\{2\}$-homog set, so $\COL(x_1,x_2,x_3)=\COL(y_1,y_2,y_3)$ iff $x_2=y_2$. 
\item
A $\{3\}$-homog set, so $\COL(x_1,x_2,x_3)=\COL(y_1,y_2,y_3)$ iff $x_3=y_3$. 
\item
A $\{1,2\}$-homog set, so $\COL(x_1,x_2,x_3)=\COL(y_1,y_2,y_3)$ iff ($x_1=y_1$ and $x_2=y_2$). 
\item
A $\{1,3\}$-homog set, so $\COL(x_1,x_2,x_3)=\COL(y_1,y_2,y_3)$ iff ($x_1=y_1$ and $x_3=y_3$). 
\item
A $\{2,3\}$-homog set, so $\COL(x_1,x_2,x_3)=\COL(y_1,y_2,y_3)$ iff ($x_2=y_2$ and $x_3=y_3$). 
\item
A $\{1,2,3\}$-homog set, so $\COL(x_1,x_2,x_3)=\COL(y_1,y_2,y_3)$ iff ($x_1=y_1$ and $x_2=y_2$ and $x_3=y_3$) (this is rainbow). 
\end{itemize} 

If $B$ is a set of three points in the plane then $\AREA(B)$ is the area of the triangle formed
by the three points in $B$. 

Let $X$ be an infinite set of points in the plane, no three colinear. 

Show that there is an infinite subset $H$ of $X$ such that the map

$B\in \binom{H}{3}$ goes to $\AREA(B)$ is injective

(So all triangles have a different area.) 


\vfill 

\centerline{\bf GO TO NEXT PAGE}

\newpage


\item
(30 points) 
Let $n\in\N$ and $n\ge 3$. 
Find the constant $\alpha,\beta\in\Q$ such that the following is true, and prove it:


If $n\equiv 0 \pmod 2$ and $\COL \colon \binom{[n]}{2} \rightarrow [2]$ then there are at least 
$\frac{n^3}{24} + \alpha n + \beta$ mono $K_3$'s. 



\vfill 

\centerline{\bf GO TO NEXT PAGE}

\newpage
\item 
(40 points) 
In this problem we guide you through another proof of the Happy ending theorem. 

\bigskip

{\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 $k$.} 

\bigskip

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
(0 points but you must do it.) 
Heya, Javier here. Bill allowed me to add a 0 point question so here it goes. This one is particularly close to my heart, so have fun with it. 
\begin{itemize}
	\item 
	If you were an animal, which species would it be? No restrictions, you can use whatever criteria you like.
	\item 
	Why did you choose that species in particular? 
\end{itemize}

\end{enumerate}

\end{document}

