In the 1980s, Ron Graham asked whether every red/blue coloring of the positive integers contains a monochromatic Pythagorean triple and offered a $100 prize for the answer. The bounty was claimed in 2016 by Heule, Kullmann, and Marek, who showed that every such coloring of [1,7825] has a monochromatic triple. Their proof used SAT solvers and took over 50,000 CPU hours to produce and verify. In this talk we will present an elementary solution to Graham’s problem that requires neither deep analytic arguments nor extensive computation.

The codegree Turán density γ(F) of a k-uniform hypergraph F is the smallest γ ≥ 0 with the following property: if every set of k−1 vertices in an n-vertex k-uniform hypergraph H lies in at least (γ+o(1))n edges, then H contains a copy of F for all sufficiently large n. Piga, Sanhueza-Matamala, and Schacht proved that this density is 1/3 for all sufficiently long 3-uniform tight cycles whose lengths are not divisible by 3. In this talk, I will discuss extensions of this result to higher uniformities, where we determine the codegree Turán density of tight cycles for a broad range of cycle lengths. I will explain the structural characterization underlying the upper bounds and the constructions used for the lower bounds. This is joint work with Jie Ma.