Given a prime power $q \equiv 1 \pmod 4$, the Paley graph of order $q$ is the graph defined over $\mathbb{F}_q$ (the finite field with $q$ elements), such that two vertices are adjacent if and only if their difference is a square in $\mathbb{F}_q$. In this talk, I will present some recent progress on the clique number of Paley graphs of non-square order, the characterization of maximum cliques in Paley graphs of square order, as well as their extensions to cyclotomic graphs. In particular, I will highlight a new proof of the Van Lint–MacWilliams’ conjecture using ideas from arithmetic combinatorics.
|
