Applications of sparse hypergraph colorings

Many problems in extremal combinatorics can be reduced to determining the independence number of a specific auxiliary hypergraph. We present two such problems, one from discrete geometry and one from hypergraph Turán theory. Using results on hypergraph colorings by Cooper-Mubayi and Li-Postle, we demonstrate that for those two problems the trivial lower bound on the independence number can be improved upon:

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  Erdős, Graham, Ruzsa and Taylor asked to determine the largest size, denoted by $g\left(n\right)$, of a subset P of the grid ${\left[n\right]}^2$ such that every pair of points in P span a different slope. Improving on a lower bound by Zhang from 1993, we show that$g\left(n\right)=\Omega \left(\frac{n^{2/3}{(\log \log n)}^{1/3}}{{\log }^{1/3}n}\right).$
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  Let $H_3^r$ denote an r-graph with $r+1$ vertices and 3 edges. Recently, Sidorenko proved the following lower bounds for the Turán density of this r-graph: $\pi \left(H_3^r\right)\ge r^{-2}$ for every r, and $\pi \left(H_3^r\right)\ge (1.7215-o(1\left)\right)r^{-2}$. We present an improved asymptotic bound:$\pi \left(H_3^r\right)=\Omega (r^{-2}{\log }^{1/2}r).$