  In 1950 Edward Nelson asked the following simple-sounding question:
  \emph{How many colors are needed to color the Euclidean plane $\mathbb{E}^2$
such that no two points distance $1$ apart are identically colored?}
  We say that $1$ is a \emph{forbidden} distance. For many years, we only knew
that the answer was $4$, $5$, $6$, or $7$. In a recent breakthrough, de Grey
\cite{degrey} proved that at least five colors are necessary.
  In this paper we consider a related problem in which we require \emph{two}
forbidden distances, $1$ and $d$. In other words, for a given positive number
$d\neq 1$, how many colors are needed to color the plane such that no two
points distance $1$ \underline{or} $d$ apart are assigned the same color? We
find several values of $d$, for which the answer to the previous question is at
least $5$. These results and graphs may be useful in constructing simpler
$5$-chromatic unit distance graphs.
