  A graphon that is defined on $[0,1]^d$ and is H\"older$(\alpha)$ continuous
for some $d\ge2$ and $\alpha\in(0,1]$ can be represented by a graphon on
$[0,1]$ that is H\"older$(\alpha/d)$ continuous. We give examples that show
that this reduction in smoothness to $\alpha/d$ is the best possible, for any
$d$ and $\alpha$; for $\alpha=1$, the example is a dot product graphon and
shows that the reduction is the best possible even for graphons that are
polynomials.
  A motivation for studying the smoothness of graphon functions is that this
represents a key assumption in non-parametric statistical network analysis. Our
examples show that making a smoothness assumption in a particular dimension is
not equivalent to making it in any other latent dimension.
