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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.01499 |
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| _version_ | 1866911900743237632 |
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| author | Clemen, Felix Christian |
| author_facet | Clemen, Felix Christian |
| contents | 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:
Erdős, 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 $$g(n)=Ω\left( \frac{n^{2/3} (\log \log n)^{1/3} }{ \log^{1/3}n} \right).$$
Let $H^r_3$ denote an $r$-graph with $r+1$ vertices and $3$ edges. Recently, Sidorenko proved the following lower bounds for the Turán density of this $r$-graph: $π(H^r_3)\geq r^{-2}$ for every $r$, and $π(H^r_3)\geq (1.7215 - o(1)) r^{-2}$. We present an improved asymptotic bound: $π(H^r_3)=Ω\left(r^{-2} \log^{1/2} r \right).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01499 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Applications of Sparse Hypergraph Colorings Clemen, Felix Christian Combinatorics 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: Erdős, 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 $$g(n)=Ω\left( \frac{n^{2/3} (\log \log n)^{1/3} }{ \log^{1/3}n} \right).$$ Let $H^r_3$ denote an $r$-graph with $r+1$ vertices and $3$ edges. Recently, Sidorenko proved the following lower bounds for the Turán density of this $r$-graph: $π(H^r_3)\geq r^{-2}$ for every $r$, and $π(H^r_3)\geq (1.7215 - o(1)) r^{-2}$. We present an improved asymptotic bound: $π(H^r_3)=Ω\left(r^{-2} \log^{1/2} r \right).$ |
| title | Applications of Sparse Hypergraph Colorings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.01499 |