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\begin{document}




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\colorlet{newred}{red!60!black}

\frame{
\frametitle{\bf BILL, RECORD LECTURE!!!!}
BILL RECORD LECTURE!!!
}

\title{\bf Introduction to the La-Lb Problem} 

\author{
\bf Exposition by William Gasarch-U of MD
}

\date{}

\maketitle

\begin{frame}\frametitle{\bf Chromatic Number of the Plane}

\pause
\smallskip
\isred{Thm} $\forall$ $\COL\colon \reals^2 \into [2]$  $\exists$ 2 points, same color, 1 inch apart. 

\pause
\smallskip
We rephrase this but first need some definitions. 

\pause 
\isblue{Def} 

\pause 
1) $\el_2$ is 2 points in the plane an inch apart. 

\pause
2) $\el_3$ is three colinear points $p_1,p_2,p_3$ where $d(p_1,p_2)=d(p_2,p_3)=1$. 

\pause 
3) You can define $\el_k$. 

\pause
4) Given a $\COL\colon\reals^2\into [2]$ a 
\isred{Red $\bm{\el_k}$} 
is an $\el_k$ where all the points in it are 
\isred{Red}. Similar for a
\isblue{Blue $\bm{\el_k}$} 

\pause 
And now the restatement

\pause
\smallskip
\isred{Thm} $\forall$ $\COL\colon \reals^2 \into [2]$  $\exists$ either a \isred{Red $\el_2$} or a
\isblue{Blue $\el_2$}. 


\end{frame}



\begin{frame}\frametitle{\bf Question}

Is the following true:

\pause
$\forall$ $\COL\colon \reals^2 \into [2]$ 
$\exists$ either a \isred{Red $\el_2$} or a
\isblue{Blue $\el_3$}. 

\isblue{Thm}
$\forall$ $\COL\colon \reals^2 \into [2]$ 
$\exists$ either a \isred{Red $\el_2$} or a
\isblue{Blue $\el_3$}. 

\pause
\smallskip
Let $\COL\colon \reals^2\into[2]$. 

\pause
\smallskip
\isblue{Case 1} There exists a \isblue{Blue $\el_3$}. \pause Then done. 

\pause
\smallskip
\isblue{Case 2} There is no \isblue{Blue $\el_3$}.

\pause
\smallskip
Hence for all points $(x,y)\in\reals^2$, 

\pause 
at least one of $(x,y)$, $(x,y+1)$, $(x,y+2)$ is $\R$. 

\pause
\smallskip
We define $\COL' \colon\reals^2\into [3]$ as follows:

\pause
\smallskip
$\COL'(x,y)$ is the least $i\in\{0,1,2\}$ such that $\COL(x,y+i)=\R$. 

\pause 
\smallskip
This is well defined because of the case we are in. 

\end{frame} 


\begin{frame}\frametitle{\bf Use That Chromatic Number of Plane is $\mathbf{\le 3}$}

\pause
$\COL'(x,y)$ is the least $i\in\{0,1,2\}$ such that $\COL(x,y+i)=\R$. 

\pause 
\smallskip
Chrom number of plane is $\le 3$, so $\exists$ $(x_1,y_1)$, $(x_2,y_2)$ such that 

\pause 
\smallskip
1) $d((x_1,y_1),(x_2,y_2))=1$, and 

\pause
\smallskip
2) $\COL'(x_1,y_1)=\COL'(x_2,y_2)=i$. 

\pause
\smallskip
Hence $\COL(x_1,y_1+i)=\R$  and $\COL(x_2,y_2+i)=\R$

\pause
\smallskip
Since $d((x_1,y_1),(x_2,y_2))=1$, $d((x_1,y_1+i),(x_2,y_2+i))=1$,

\pause
\smallskip
So $(x_1,y_1+1)$ and $(x_2,y_2+1)$ are a \isred{Red $\el_2$}.  \pause Done!

\end{frame} 

\begin{frame}\frametitle{\bf Can Prove Result About $\mathbf{(\el_2,\el_4)}$}

Using that the Chromatic Number of the Plane ($\chi$) is $\le 4$ one can easily prove
the following:

\isblue{Thm}
$\forall$ $\COL\colon \reals^2 \into [2]$ 
$\exists$ either a \isred{Red $\el_2$} or a
\isblue{Blue $\el_4$}. 

\pause
\smallskip
Juhasz prove the above theorem without using $\chi\le 4$.
In fact, he prove the theorem in 1979, 39 years before $\chi\le 4$ was proven.

\end{frame} 

\begin{frame}\frametitle{\bf Notation}

\isblue{Notation} Let $a,b\ge 2$.  $\reals^2 \into (\el_n,\el_m)$ means

\pause
$\forall  \COL \colon \reals^2 \into [2]$ $\exists$ \isred{Red $\el_n$}  or \isblue{Blue $\el_m$}.

\pause
\smallskip
We proved $\reals^2 \into (\el_2,\el_3)$.

\pause
\smallskip
We will soon present statements about $\reals^2\into (\el_2,\el_n)$

\end{frame} 

\begin{frame}\frametitle{\bf Notes About the Statements}

\pause
The next two slides have statements about what is known. 

\pause
\smallskip
All of the papers referred to are on this website: 

\url{https://www.cs.umd.edu/~gasarch/TOPICS/ERT/ert.html}
\end{frame} 

\begin{frame}\frametitle{\bf $\mathbf{\reals^2\into (\el_2,\el_n)}$?}

\begin{tabular}{|c|c|c|}
\hline
Author and Year & Result & About $\reals^2$\cr
\hline
\isblue{Positive Results} & & \cr
\hline
\pause 
 Szlam, 1999  &    $(\forall X)[|X|=3]$ &  \cr
              &    $\reals^2\into (\el_2,X)$ & $\reals^2\into (\el_2,\el_3)$ \cr
\hline
\pause 
Juhasz, 1979 &    $(\forall X)[|X|=4]$ &       \cr
             &    $\reals^2\into (\el_2,X)$ & $\reals^2\into (\el_2,\el_4)$\cr
\hline
\pause 
Tsaturian, 2017 &    $\reals^2\into (\el_2,\el_5)$ & $\reals^2\into (\el_2,\el_5)$ \cr
\hline
\pause 
Arman-Tsaturian, 2017 & $\reals^3\into (\el_2,\el_{6})$ & NONE \cr
\hline
\pause 
\isred{Negative Results} & & \cr
\hline
\pause 
Csizmadia-Togh 1994 & $\exists X\subseteq \reals^2$, $|X|=8$&  \cr 
                    & $\reals^2\not\into (\el_2,X)$ & NONE \cr 
\hline
\pause 
Conlon-Fox, 2018    & $(\forall n\ge 2)$            &      \cr 
                    & $\reals^n\not\into (\el_2,\el_{10^{10}})$ & $\reals^2\not\into(\el_2,\el_{10^{50}})$ \cr 
\hline
\pause
Currier et al., 2026 & $\reals^2\not\into (\el_2,\el_{6330})$ & $\reals^2\not\into (\el_2,\el_{6330})$ \cr
\hline
\pause
Hu, 2026 & $\reals^2\not\into (\el_2,\el_{175})$ & $\reals^2\not\into (\el_2,\el_{175})$ \cr
\hline
\end{tabular} 

\pause
\medskip
\isblue{Open} 
Narrow the gap between $\reals^2\into (\el_2,\el_5)$ and $\reals^2\not\into (\el_2,\el_{175})$. 

\end{frame} 

\begin{frame}\frametitle{\bf $\mathbf{\reals^2\into (\el_3,\el_n)}$?}

All of the negative results hold for all $n\ge 2$. Our interest is in the
$n=2$ case. 

\begin{tabular}{|c|c|}
\hline
Author and Year & Result \cr 
\hline
\isblue{Positive Results} &  \cr
\pause 
Currier-Moore-Yip, 2024&    $\reals^2\into (\el_3,\el_3)$ \cr
\hline
\isred{Negative Results} & \cr
\pause 
Conlon-Wu, 2022 & $\reals^n\not\into (\el_3,\el_{10^{50}})$ \cr 
\pause 
Fuhrer-Toth, 2024 & $\reals^n\not\into (\el_3,\el_{1177})$ \cr 
\pause 
Currier-Moore-Yip, 2024& $\reals^n\not\into (\el_3,\el_{20})$  \cr 
                       & $\reals^n\not\into (\el_4,\el_{18})$ \cr 
                       & $\reals^n\not\into (\el_5,\el_{10})$ \cr 
\hline
\end{tabular} 

\pause
\bigskip
\isblue{Open Problems} 

\pause
\medskip
Narrow the gap between $\reals^2\into (\el_3,\el_3)$ and $\reals^2\not\into (\el_3,\el_{20})$.

\pause
\medskip
The results of Fuhrer-Togh and Currie-Moore-Yip are messy. 
For some reasonable values of $n$ find nice proofs that 
$\reals^2\not\into (\el_3,\el_n)$. 

\end{frame} 

\begin{frame}\frametitle{\bf One More Result}
\begin{tabular}{|c|c|c|}
\hline
Author and Year & Result  \cr 
\hline
\isred{Negative Results} &  \cr
Erdos et al. See Below& $\reals_2\not\into (\el_6,\el_6)$ \cr 
\hline
\end{tabular} 

\pause
\smallskip
\Erdosns, Graham, Montgomery, Rothchild, Spencer, Strauss.

\pause
\smallskip
The title of the paper is \isblue{Euclidean Ramsey Theorems I (1973)} and was 
(obviously) an early paper on Euclidean Ramsey Theory. 

\end{frame}

\begin{frame}\frametitle{\bf Open Problems about $\mathbf{\el_2}$, $\mathbf{\el_3}$}

\isblue{$\mathbf{\el_2}$}

\pause 
\isred{Known} $\reals^2\into (\el_2,\el_5)$

\pause 
\isred{Known} $\reals^2\not\into (\el_2,\el_{175})$

\pause 
\isred{Open} Narrow the Gap. 

\bigskip
\pause 
\isblue{$\mathbf{\el_3}$}

\pause 
\isred{Known} $\reals^2\into (\el_3,\el_3)$

\pause 
\isred{Known} $\reals^2\not\into (\el_3,\el_{20})$

\pause 
\isred{Open} Narrow the Gap. 

\end{frame}

\begin{frame}\frametitle{\bf Open Problems about $\mathbf{\el_4}$, $\mathbf{\el_5}$}

\isblue{$\mathbf{\el_4}$}

\pause 
\isred{Known (not)} Do not know any $a\ge 4$ with $\reals^2\into (\el_4,\el_a)$

\pause 
\isred{Known} $\reals^2\not\into (\el_4,\el_{18})$

\pause 
\isred{Open} Narrow the Gap. 

\pause
Might be $\reals^2\not\into (\el_4,\el_4)$ so there is no such $a$. 

\bigskip
\pause 
\isblue{$\mathbf{\el_5}$}

\pause 
\isred{Known (not)} Do not know any $a\ge 5$ with $\reals^2\into (\el_5,\el_a)$

\pause 
\isred{Known} $\reals^2\not\into (\el_5,\el_{10})$

\pause 
\isred{Open} Narrow the Gap. 

\pause
Might be $\reals^2\not\into (\el_5,\el_5)$ so there is no such $a$. 

\bigskip
\pause 
\isblue{$\mathbf{\el_6}$}

\pause 
\isred{Known} $\reals^2\not\into (\el_6,\el_6)$.

\pause
No Open Problem.
\end{frame}

\begin{frame}\frametitle{\bf General Open Problems}

\pause
The papers with large numbers have nice proof since they just use numbers that work.

\pause
\smallskip
Some of the papers with small numbers are messy and rather particular.

\pause 
\smallskip
$\reals^2\into (\el_3,\el_3)$ is one of them

\pause
\smallskip
\isred{Open} Get nicer proofs.

\pause
\smallskip
\isred{Open} In some cases get nicer proofs of weaker results.

\end{frame}


\begin{frame}\frametitle{\bf What Do People Think Is True?} 

I emailed these slides to Conlon and Fox to ask 

\pause
Any results on  $\reals^n\into$ or $\reals^n\not\into$ absent? (NO.) 

\pause
\smallskip
What does the good money say is true of the open problems?

\pause
\smallskip
Fox's answer surprised me:

\pause
\smallskip
\isblue{Some of these problems may be independent of ZF}

\pause
\smallskip
\isblue{My Opinion} This would be both awesome and awful.

\pause 
\isblue{Awesome} Natural Problems Ind of ZF. Cool!

\pause 
\isblue{Awful} I went answers! Dagnabbit! 

\pause
\isblue{Caveat} If we assume choice these questions have answers.

\pause
\isblue{What Does Ind of ZF Mean?} Proof will be hard and unlike known papers
all of which are in ZF. 

\pause
\isblue{Conjectures?} None.

\end{frame}

\end{document} 
