That is the long title of an article which is joint work with Francesco Pavese. Last week, I received the news that it has been accepted for publication, my very first publication! In the article, we continue some work that was started in an earlier preprint with Leo Storme, which ironically, is still under review.

I have posted about this topic in an earlier blog post, but I’ll give a quick recap here.

Roughly speaking the question is the following: consider a finite projective plane of order and a unitary polarity of this plane (see [1] for all about this). A polar triangle with respect to is a triple of points such that for any triple of distinct indices. We are interested in large sets of points such that no triple from this set forms a polar triangle. This is a particular instance of the classical forbidden configuration problem, which has appeared in many forms in the realm of extremal combinatorics.

Using a combination of algebraic graph theory techniques and geometrical constructions, we manage to obtain upper bounds for general projective planes, and lower bounds for the Desarguesian and Figueroa plane, which asymptotically match! It was the first time I got to use eigenvalue interlacing [2] in a new environment, so I’m quite content with the result.

It has been accepted to the European Journal of Combinatorics, but until its actual publication, you can check it out on the arXiv.

**References**

[1] D. Hughes, F. Piper, *Projective planes*, Graduate Texts in Mathematics, Springer-Verlag New York-Berlin, 1973.

[2] W.H. Haemers, *Interlacing Eigenvalues and Graphs*, Linear Algebra Appl.,

226/228:593–616, 1995.