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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.