Abstract Given a finite abelian group G and a subset $$J\subset G$$ J ⊂ G with $$0\in J$$ 0 ∈ J , let $$D_{G}(J,N)$$ D G ( J , N ) be the maximum size of $$A\subset G^{N}$$ A ⊂ G N such that the difference set $$A-A$$ A - A and $$J^{N}$$ J N have no non-trivial intersection. Recently, this extremal problem has been widely studied for different groups G and subsets J . In this paper, we generalize and improve the relevant results by Alon and by Hegedűs by building a bridge between this problem and cyclotomic polynomials with the help of algebraic graph theory. In particular, we construct infinitely many non-trivial families of G and J for which the current known upper bounds on $$D_{G}(J, N)$$ D G ( J , N ) can be improved exponentially.

Abstract Given two irreducible conics C and D over a finite field $$\mathbb {F}_q$$ F q with q odd, we show that there are $$q^2/4+O(q^{3/2})$$ q 2 / 4 + O ( q 3 / 2 ) points P in $$\mathbb {P}^2(\mathbb {F}_q)$$ P 2 ( F q ) such that P is external to C and internal to D. This answers a question of Korchmáros. We also prove the analogous result for higher-dimensional smooth quadric hypersurfaces in $$\mathbb {P}^{n-1}$$ P n - 1 with n odd, where the answer is $$q^{n-1}/4+O(q^{n-\frac{3}{2}})$$ q n - 1 / 4 + O ( q n - 3 2 ) .

Abstract Let q be an odd power of a prime p , and $$S \subseteq \mathbb {F}_q^*$$ S ⊆ F q ∗ such that $$S=-S$$ S = - S and $$S/S \ne \mathbb {F}_q^*$$ S / S ≠ F q ∗ . We show that the clique number of the Cayley graph $$\operatorname {Cay}(\mathbb {F}_q^+,S)$$ Cay ( F q + , S ) is at most $$\sqrt{|S/S|}+\sqrt{q/p}$$ | S / S | + q / p , improving the best-known $$\sqrt{q}$$ q upper bound for many families of such graphs substantially. Such a new bound is strongest for cyclotomic graphs and in particular, it implies the first nontrivial upper bound on the clique number of all generalized Paley graphs of non-square order, extending the work of Hanson and Petridis. Moreover, our new bound is asymptotically sharp for an infinite family of generalized Paley graphs, and we further discover the first nontrivial family among them for which the clique number can be exactly determined. We also obtain a new lower bound on the number of directions determined by a large Cartesian product in the affine Galois plane AG (2, q ), which is sharp for infinite families.