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\begin{document}

\centerline{\bf CMSC 752 Homework 10} 
\centerline{\bf Morally Due Tue April 15, 2025} 
\centerline{\bf Dead Cat April 17}

\bigskip

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

\smallskip

\begin{enumerate} % 1

\item 
(50 points) 

{\bf Motivation} As every Hungarian Kindergarten child knows:

\begin{enumerate} %2
\item
$\exists$ $\COL\colon \binom{[5]}{2} \into [2]$ with NO homog sets.
Note that this is the worse case. We wonder about what USUALLY happens.
\item
For every $\COL\colon \binom{[6]}{2} \into [2]$ there exists two homog sets.
Note that this is the worse case. We wonder about what USUALLY happens.
\end{enumerate} %2

Problem is on the next page

\newpage

\begin{enumerate} %2
\item
(Nothing to hand in for this step.) 
Write a program that will, on input $n\in\N$
generate a coloring of the edges of $K_n$  at random: for each edge,
prob of RED is $\frac{1}{2}$ and prob of BLUE is 
$\frac{1}{2}$. 
Use adjacency matrices for the graph. 


\item
(Nothing to hand in for this step.) 
Write a program that will, on input a graph $K_n$ that has its edges 2-colored, 
determines {\it how many}  Homog sets of size 3 are there.

\item
(Nothing to hand in for this step.) 
Write the following program. 

1) Input $n$. 

2) For $i=1$ to 100
\begin{enumerate} %3
\item 
Generate a graph using the program in Part 1. 
\item
Find how many homog triangles there are using the program in Part 2.
\item
Let $A[i]$ be the number of homog triangles.
\end{enumerate}  %3

\item
Calculate the MIN, MAX, and MEAN of the $A[i]$'s.

\item 
(This you hand in.) 

Write a program that generates the following table
(I have made up the numbers). 

\begin{tabular}{|c|c|c|c|}
\hline
n        & MIN      & MAX        &  MEAN  \cr
\hline
5        & 0        & 5          &  3        \cr
6        & 0        & 5          &  3        \cr
$\vdots$ & $\vdots$ & $\vdots$   & $\vdots$  \cr
40       & 0        & 5          &  2        \cr
\hline
\end{tabular}


\item
In class we showed that in the worst case there will be
$\sim \frac{1}{4}\binom{n}{3}$ mono triangles. 
Find $A,B,C$ such that the following seems to be true empirically:

{\it If you choose a coloring at random then 

\begin{itemize}
\item
The MIN will be $\sim A\binom{[n]}{3}$.
\item
The MAX will be $\sim B\binom{[n]}{3}$.
\item
The AVG will be $\sim C\binom{[n]}{3}$.
\end{itemize} 
}

\end{enumerate} %2

\newpage

\item
(50 points)
Prove  the following statement 

{\it For all $c$ there exists a finite set of grids $\OBS_c$ such that 

$n\times m$ is $c$-colorable iff $n\times m$ does not contain any element of $\OBS_c$. 
}

\item
(Extra credit)

\begin{enumerate} 
\item Give your name.
\item 
Use Spencer's proof to find reasonably sized graphs $G=(V,E)$ such that
\begin{itemize} 
\item 
$K_4$ is not a subgraph of $G$, and
\item
For all $\COL \colon E\into [2]$ there exists a mono triangle. 
\end{itemize} 
\end{enumerate} 


\end{enumerate} %1


\end{document}