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\def\course{MATH 314}
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\def\assignment{HW \#13}
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\def\due{Due on May 2}




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\begin{document}
\noindent  Unless explicitly requested by a problem, do not include sketches as part of your proof. You are free to use the result from any problem on this (or previous) assignment as a part of your solution to a different problem even if you have not solved the former problem.
\vskip10pt

\begin{problem}[2pts]
	Prove that $R(3,n)\leq n^2$ for every positive integer $n$.
\end{problem}


\begin{problem}[2pts]
	The $3$-color Ramsey number $R(m,n,p)$ is the smallest integer $N$ such that every 3-coloring of $E(K_N)$ (say with colors red, blue and green) contains either a red copy of $K_m$, a blue copy of $K_n$ or a green copy of $K_p$.
	\vskip5pt
	
	Prove that $R(3,3,3)\leq 17$.
\end{problem}


\begin{problem}[2pts]
Suppose that $G$ is a graph that has no induced $K_{1,4}$ and such that $\omega(G) = 4$. Show that $\Delta(G)\leq 17$.

Hint: $17+1=R(4,4)$.
\end{problem}


\begin{problem}[2pts]
		Show that $N=7$ is the smallest integer $N$ such that every red,blue-coloring of $E(K_N)$ contains either a red $K_{1,3}$ or a blue copy of $K_3$.
\end{problem}


\begin{problem}[2 pts]
	Let $n$ be a positive integer and set $N=3n-1$.
		Construct a red,blue-coloring of $E(K_{N-1})$ which \emph{does not} contain a monochromatic matching with $n$ edges.
		
		Hint: Break $V(K_{N-1})$ into two pieces, one of size $2n-1$ and the other of size $n-1$, and color the edges based on how they intersect these two pieces.
\end{problem}


\end{document}