Applications of Sparse Hypergraph Colorings

TL;DR

This paper explores applications of sparse hypergraph colorings to improve bounds in extremal combinatorics problems, including grid point sets with unique slopes and Turán densities of specific hypergraphs, demonstrating novel lower bounds.

Contribution

The paper introduces new lower bounds for independence numbers and Turán densities using hypergraph coloring results, advancing understanding in extremal combinatorics.

Findings

Improved lower bound for the size of point sets with unique slopes: g(n)=Ω(n^{2/3} (log log n)^{1/3} / log^{1/3} n).

Enhanced asymptotic bound for Turán density of H^r_3: Ω(r^{-2} log^{1/2} r).

Demonstrates the application of hypergraph coloring results to specific combinatorial problems.

Abstract

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 Tur\'an 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: Erd\H{o}s, Graham, Ruzsa and Taylor asked to determine the largest size, denoted by $g(n)$, of a subset $P$ of the grid $[n]^2$ such that every pair of points in $P$ span a different slope. Improving on a lower bound by Zhang from 1993, we show that $$g(n)=\Omega \left( \frac{n^{2/3} (\log \log n)^{1/3} }{ \log^{1/3}n} \right).$$ Let $H^r_3$ denote an $r$-graph with $r+1$ vertices and $3$ edges. Recently, Sidorenko proved the following lower bounds for the Tur\'an density of this $r$-graph: $\pi(H^r_3)\geq r^{-2}$ for every $r$, and $\pi(H^r_3)\geq (1.7215 - o(1)) r^{-2}$. We present an improved asymptotic bound: $\pi(H^r_3)=\Omega\left(r^{-2} \log^{1/2} r \right).$