# Graham, Ronald L.

## Termination of the sixth Goodstein Sequence ★

Author(s): Graham

\begin{question} How many steps does it take the sixth Goodstein sequence to terminate? \end{question}

Keywords: Goodstein Sequence

## Monotone 4-term Arithmetic Progressions ★★

Author(s): Davis; Entringer; Graham; Simmons

\begin{question} Is it true that every permutation of positive integers must contain monotone 4-term arithmetic progressions? \end{conjecture}

Keywords: monotone arithmetic progression; permutation

## Pebbling a cartesian product ★★★

Author(s): Graham

We let $p(G)$ denote the pebbling number of a graph $G$.

\begin{conjecture} $p(G_1 \Box G_2) \le p(G_1) p(G_2)$. \end{conjecture}

## Divisibility of central binomial coefficients ★★

Author(s): Graham

\begin{problem}[1] Prove that there exist infinitely many positive integers $n$ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$$ \end{problem}

\begin{problem}[2] Prove that there exists only a finite number of positive integers $n$ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7\cdot 11) = 1.$$ \end{problem}

Keywords:

## The large sets conjecture ★★★

Author(s): Brown; Graham; Landman

\begin{conjecture} If $A$ is 2-large, then $A$ is large. \end{conjecture}

Keywords: 2-large sets; large sets

## Graham's conjecture on tree reconstruction ★★

Author(s): Graham

\begin{problem} for every graph $G$, we let $L(G)$ denote the \Def{line graph} of $G$. Given that $G$ is a tree, can we determine it from the integer sequence $|V(G)|, |V(L(G))|, |V(L(L(G)))|, \ldots$? \end{problem}

Keywords: reconstruction; tree