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Finding a reasonably good upper bound for the clique number of Paley graphs is an open problem in additive combinatorics. A recent breakthrough by Hanson and Petridis using Stepanov's method gives an improved upper bound on Paley graphs…

We derive an asymptotic formula for the number of solutions in a given subfield to certain system of equations over finite fields. As an application, we construct new families of maximal cliques in generalized Paley graphs. Given integers…

Let $q$ be an odd power of a prime $p$, and $S \subset \mathbb{F}_q^*$ such that $S=-S$ and $S/S \neq \mathbb{F}_q^*$. We show that the clique number of the Cayley graph $\operatorname{Cay}(\mathbb{F}_q^+,S)$ is at most…

Let $n=2^s p_{1}^{\alpha_{1}}\cdots p_{k}^{\alpha_{k}}$, where $s=0$ or $1$, $\alpha_i\geq 1$, and the distinct primes $p_i$ satisfy $p_i\equiv 1\pmod{4}$ for all $i=1, \ldots, k$. Let $\mathbb{Z}_n^\ast$ denote the group of units in the…

Let $GP(q,d)$ be the $d$-Paley graph defined on the finite field $\mathbb{F}_q$. It is notoriously difficult to improve the trivial upper bound $\sqrt{q}$ on the clique number of $GP(q,d)$. In this paper, we investigate the connection…

This work addresses the block-diagonal semidefinite program (SDP) relaxations for the clique number of the Paley graphs. The size of the maximal clique (clique number) of a graph is a classic NP-complete problem; a Paley graph is a…

Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise…

We describe a new class of maximal cliques, with a vector space structure, of Cayley graphs defined on the additive group of a field. In particular, we show that in the cubic Paley graph with order $q^3$, the subfield with $q$ elements…

There is a conjecture that the second largest maximal cliques in Paley graphs of square order $P(q^2)$ have size $\frac{q+\epsilon}{2}$, where $q \equiv \epsilon \pmod 4$, and split into two orbits under the full group of automorphisms…

We prove that the degree 4 sum-of-squares (SOS) relaxation of the clique number of the Paley graph on a prime number $p$ of vertices has value at least $\Omega(p^{1/3})$. This is in contrast to the widely believed conjecture that the actual…

It is known that the number of directions formed by a Cartesian product $A \times B \subset AG(2,p)$ is at least $|A||B| - \min\{|A|,|B|\} + 2$, provided $p$ is prime and $|A||B|<p$. This implies the best known upper bound on the clique…

We give a new unified proof that any simple graph on $n$ vertices with maximum degree at most $\Delta$ has no more than $a\binom{\Delta+1}{t}+\binom{b}{t}$ cliques of size $t \ (t \ge 3)$, where $n = a(\Delta+1)+b \ (0 \le b \le \Delta)$.

For a prime $p$ we define the Paley graph to be the graph with the set of vertices $\mathbb{Z}/p\mathbb{Z}$, and with edges connecting vertices whose sum is a quadratic residue. Paley graphs are notoriously difficult to study, particularly…

We show the maximality of subfields as cliques in a special family of Cayley graphs defined on the additive group of a finite field. In particular, this confirms a conjecture of Yip on generalized Paley graphs.

In 1994 S. McGuinness showed that any greedy clique decompo- sition of an n-vertex graph has at most $\lfloor n^2/4 \rfloor$ cliques (The greedy clique decomposition of a graph, J. Graph Theory 18 (1994) 427-430), where a clique…

Finding the largest clique is a notoriously hard problem, even on random graphs. It is known that the clique number of a random graph G(n,1/2) is almost surely either k or k+1, where k = 2log n - 2log(log n) - 1. However, a simple greedy…

Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results: \medskip \noindent $\bullet$ If the bicliques partition the edges…

A well-known conjecture in analytic number theory states that for every pair of sets $X,Y\subset\mathbb{Z}/p\mathbb{Z}$, each of size at least $\log ^C p$ (for some constant $C$) we have that the number of pairs $(x,y)\in X\times Y$ such…

We study subgraphs of Paley graphs of prime order $p$ induced on the sets of vertices extending a given independent set of size $a$ to a larger independent set. Using a sufficient condition proved in the author's recent companion work, we…

Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best…