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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 knewthat 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 twopoints distance $1$ \underline{or} $d$ apart are assigned the same color? Wefind several values of $d$, for which the answer to the previous question is atleast $5$. These results and graphs may be useful in constructing simpler$5$-chromatic unit distance graphs.

The Hadwiger-Nelson problem with two forbidden distances