= \frac{\ell + \sqrt{\ell(\ell-e(\ell+1))}}{\ell(\ell+1)}.$$ There is a partition of $\{1,\ldots,n\}$ into $\ell$ pieces, the first $\ell-1$ of size $\lfloor cn\rfloor$ and the last of size between $\lfloor cn\rfloor$ and $2\lfloor cn\rfloor$, such that $G$ is the complete $\ell$-partite graph on these pieces, plus a number of additional edges within the last piece. ($\lfloor y\rfloor$ denotes the largest integer greater than or equal to $y$.) These additional edges can take any form, as long as there are no triangles within the last piece.

This means that, after possibly renumbering the vertices, the graphon for such a graph can be written as an uneven $\ell \times \ell$ checkerboard obtained from cutting the unit interval into pieces $V_k=[(k-1)c,kc]$ for $k<\ell$ and $V_\ell=[(\ell-1)c, 1]$, with the checkerboard being 1 outside the main diagonal, 0 on the main diagonal except the upper right corner, and corresponding to a zero-triangle graph in the upper right corner.

Limits of such graphons in the metric must take the form $$g(x,y) = \begin{cases} 1 & x< kc < y \hbox{ or } y<kc < x \hbox{ for an integer }k<\ell; \cr 
0 & (k-1) c < x,y < kc \hbox{ for some integer }k<\ell; \cr
\hbox{unspecified} & x,y > 