I need assistance with the following problem:
Let $S$ be a set of $n$ points in the plane. Prove that there is $O(n^{7/3})$ triangles of unit perimeter among $S$.
Let $\mathcal{E}$ be the set of ellipses formed by taking every pair of points $a, b, \in S$ as foci and sum of distances from foci equal to $1 - |a-b|$. Clearly, this is a problem involving incidences between $\mathcal{E}$ and incidences. I tried applying the crossing lemma, but this can only be applied on a simple graph. Consider the graph where vertices are $S$ and edges are between adjacent nodes on an ellipse. There are $q$ edges, hence $q$ incidences. We can remove all loops and double edges between nodes by removing those ellipses. After removal, this gives us at least $q - 2 \binom{n}{2}$ edges. But now I do not know how to bound the multiplicity.
Comments, suggestions, solutions are appreciated.
