I started writing this blog post some months ago. Occasion was that my paper “A construction for clique-free pseudorandom graphs” (in joint work Anurag Bishnoi and Valentina Pepe) was accepted by Combinatorica with minor revisions. More precisely, one of the referees was unfavorable of publication because he got the impressions that we are simply restating a result by Bannai, Hao and Song. I think that the referee had a point, but for slightly wrong reasons. This triggered me to do two things. First of all, it made me include more history of the construction in our actual paper. Then I wanted to write a blog post about the history of the construction. Sadly, I wanted to include too much history in my first attempt to write this post, so it was very much out of scope. Here now a more concise version of my original plan.
Let be the orthogonality graph, that is the graph with as vertices with two vertices adjacent if they are orthogonal. So are adjacent if . There are many publications which investigate this problem. The aim of this post is two fold:
- To summarize the state of the art.
- To demonstrate how careful literature study is helpful to obtain results.
During this post I avoid using many definition, while I cite things that use all kind of terminology. For this I wrote this blog post as a reference. We will go through the main steps in the paper and focus more on the wider context of the research, so the parts for which there is less space in papers and not enough time in talks.