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\begin{document}

\centerline{\bf 752 Final for Spring 2026}

\bigskip

{\bf Exam START TIME:} 12:00PM (Noon) May 15

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{\bf Exam END TIME:}    6:00PM May 15

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{\bf My Expectation:} The exam should take between 2 and 3 hours,
but feel free to take all 6.

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{\bf Rules:} 

\begin{itemize}
\item 
You CAN use your slides, notes, HW, and your brain. 
\item
You CAN ask me a question of clarity by email. I may also be on zoom
part of the day. 
\item
You CANNOT use anything else.
\item
How will I enforce the rules? I am counting on your integrity and fine character. 
\item
The exam is one 0-point problem (what is your name) and then 
four 20-point problems. So the  totals to 80 points. The other 20 points are from the 
take-home part of the final. 
\item
Exam starts on the next page.
\end{itemize}



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\item
(0 points)
What is your name


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\item 
If $X$ is a finite subset of $\N$ then $\min(X)$ is the smallest
element of $X$. 

Recall that $\ACK$ is a very fast growing function of two variables. 

We want a fast growing function on one variable.

Let $f(n)=\ACK(n,n)$. 

Note that $f$ is a fast growing function on one variable.


\bigskip

{\bf Def} Let $X$ be a finite subset of $\N$. $X$ is {\it superlarge} if $|X|> f(\min(X))$. 

Prove the following: 

{\it For every $k\in\N$ there exists $n$ such that, 

for every $\COL\colon \binom{k,k+1,\ldots,k+n}{3}\into [2]$ there exists a superlarge homog set.} 

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\item
(20 points) 
Recall that infinite $a$-ary Ramsey Theorem:

{\it For all $a\in\N$, for all $\COL\colon \binom{\N}{a}\into[2]$ 
there exists an infinite homog set.}

We denote the above statement as $a$-ary IRT (Infinite Ramsey Theorem). 

Assume that you already know the 1-ary IRT, 2-ary IRT, $\ldots$, 99-ary IRT.

Prove the 100-ary IRT. 

(Hint: There are many different proofs depending on which $a$-IRT you
want to use infinitely often and which one at the end. Do the easiest one
which is NOT the one that gives the best bounds when made finite.) 


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\item
(20 points)
Let $E$ be the equation

$$x_1 + 2x_2 + 4x_3 - 8x_4 = 0$$

Give a finite coloring of $\N$ that has no mono solution in $\N$.
Prove your coloring has no mono solution. 
(For this problem 0 is NOT an element of $\N$.) 

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\item
(20 points)
Recall that $W(k,2)$ is 

{\it the least $n$ such that, for all $\COL\colon [n]\into [2]$, there is a mono $k$-AP.}

Use the probabilistic method to obtain a lower bound on $W(k,2)$. 


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