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\centerline{\bf CMSC 752 Homework 8} 
\centerline{\bf Morally Due Tue April 1, 2025} 
\centerline{\bf Dead Cat April 3}

\bigskip

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

\smallskip

{\bf IMPORTANT} In this HW when we refer to {\it Coloring $K_n$} we mean coloring
the EDGES of $K_n$. 

\smallskip

{\bf IMPORTANT} In this HW when we ask for a table of numbers or for a number, if its a REAL
number we want it to just 2 places. So for us EVERY DAY IS $\pi$-DAY, since we would use 3.14
for $\pi$.

\smallskip

\begin{enumerate}

\item 
(50 points) 
In class we proved the following two theorems. 

{\bf GP Theorem:}
$\forall m\ge 1$
$\forall $ 2-col of $K_{f(m)}$ $\exists$ $m$ mono $K_4$'s

where $f(m)=m+17$. 

{\bf ST Theorem:} 
$\forall m\ge 2$
$\forall$ 2-col of $K_{g(m)}$ $\exists$ $m$ mono $K_4$'s

where 
$g(m)$ be the least $n$ such that 
$$n\times (n-1)\times (n-2)\times (n-3) > 73440 (m-1).$$

Note that $g(m)\sim m^{1/4}$. 

{\bf Ratio Version} 
$\forall $ 2-col of $K_{n}$, $\exists$ $\ge \frac{1}{3060}\binom{n}{4}$ mono $K_4$'s. 
(We won't need the Ratio Version for this problem, but we will have an analog of it
in Problem 2, and it will be discussed in the Extra Credit Problem.)

\begin{enumerate}
\item
Make a table with five columns: $m$, $f(m)$, $g(m)$, $m^{1/4}$, $g(m)/m^{1/4}$,
for $m=2$ to $m=100$. 
It should look like this (the numbers are fake and I only go out two rows).

\[
\begin{array}{|c|c|c|c|c|}
\hline
m & f(m) & g(m) & m^{1/4} & g(m)/m^{1/4} \cr 
\hline
2 & 19   & 22   & 1.18 & 18.64 \cr
3 & 20   & 24   & 2.21 & 10.86 \cr
\hline
\end{array}
\]

\item
What is the least $m$ such that $g(m)<f(m)$. 

\item
Make the table up to 1000 (do not hand that in). 
Find a constant $A$ such that 
$g(m)= A m^{1/4}$ is a good approximation for $g(m)$.
(I am assuming that the last column in your table has a limit.
I have not done this problem so I could be wrong.) 

\end{enumerate}

\newpage

\item 
Fill in $W,X,Y,Z$ and prove the following Theorems.
$W$, $Y$, $Z$ are numbers. $X$ is a statement. 

{\bf GP Theorem:}
$\forall m\ge 1$
$\forall $ 2-col of $K_{f(m)}$ $\exists$ $m$ mono $K_5$'s

where $f(m)=m+W$. 

{\bf ST Theorem:} 
$\forall m\ge 2$
$\forall$ 2-col of $K_{g(m)}$ $\exists$ $m$ mono $K_5$'s

where 
$g(m)$ be the least $n$ such that $X$.

Note that $g(m)\sim m^Y$. 

{\bf Ratio Version} 
$\forall $ 2-col of $K_{n}$, $\exists$ $\ge Z\binom{n}{5}$ mono $K_5$'s. 

\begin{enumerate}
\item
Make a table with five columns: $m$, $f(m)$, $g(m)$, $m^Y$, $g(m)/m^Y$. 
for $m=2$ to $m=100$. 

\item
What is the least $m$ such that $g(m)<f(m)$.
(I have not done this problem so I do not know if $m\le 100$.)

\item
Make the table up to 1000 (do not hand that in). 
Find a constant $A$ such that 
$g(m)= A m^{Y}$ is a good approximation for $g(m)$.
(I am assuming that the last column in your table has a limit.
I have not done this problem so I could be wrong.) 
\end{enumerate}

\newpage
\item
(0 Points, Extra Credit)
\begin{enumerate}
\item
Type your name here. (This will not get you any extra credit)
\item
In class we proved

{\bf Ratio Version} 
$\forall $ 2-col of $K_{n}$ $\exists$ $\ge \frac{1}{3060}\binom{n}{4}$ mono $K_4$'s. 

\bigskip
Improve the constant from $\frac{1}{3060}$ to something larger. 
(This might be open.)
\end{enumerate}

\end{enumerate}

\end{document}