A Very Short History of Pseudorandom Cliquefree Graphs

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.

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Pseudorandom clique-free graphs

Anurag Bishnoi wrote a post about a recently finished preprint on pseudorandom clique-free graphs written by me, Anurag, and Valentina Pepe. We (slightly) improve a construction by Alon and Krivelevich from 1997.

Anurag's Math Blog

Pseudorandom graphs are graphs that in some way behaves like a random graph with the same edge density. One way in which this happens is as follows. In the random graph $latex G(n, p)$, with $latex p = p(n) leq 0.99$, a direct application of Chernoff bound implies that the probability of the following event approaches $latex 1$ as $latex n$ approaches infinity:

$latex |e(S, T) – p|S||T|| = O(sqrt{pn |S||T|}$

where $latex S,T$ are arbitrary subsets of vertices and $latex e(S,T)$ denotes the number of edges with one end vertex in $latex S$ and the other one in $latex T$.  Note that $latex p|S||T|$ is the expected number of edges between $latex S$ and $latex T$ in this model, and $latex sqrt{p(1 – p)|S||T|}$ is the standard deviation. Now let $latex G$ be a $latex d$-regular graph on $latex n$-vertices and let $latex lambda$ be the second largest…

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