Given a simple graph Γ, we describe a "lifting" to a 3-uniform hypergraph ϕ(Γ) that sends the complement of Γ to the complement of ϕ(Γ). We consider the effects of this lifting on cycles, complete subhypergraphs, and complete subhypergraphs missing a single hyperedge. Our results lead to natural lower bounds for some hypergraph Ramsey numbers.

For a fixed prime p Á 1 .mod 4/, we define the corresponding quartic residue graph and determine the number of triangles contained in such a graph. Our computation requires us to compute the number of pairs of consecutive quartic residues modulo p via the evaluation of certain quartic Jacobi sums.

The most natural extensions to the law of quadratic reciprocity are the rational reciprocity laws, described using the rational residue symbol. In this article, we provide a reciprocity law from which many of the known rational reciprocity laws may be recovered by picking appropriate primitive elements for subfields of ‫(ޑ‬ζ p ). As an example, a new generalization of Burde's law is provided.

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