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Embedding tetrahedra into quasirandom hypergraphs
- Source :
- J. Combin. Theory Ser. B 121 (2016), 229-247
- Publication Year :
- 2016
-
Abstract
- We investigate extremal problems for quasirandom hypergraphs. We say that a $3$-uniform hypergraph $H=(V,E)$ is $(d,\eta)$-quasirandom if for any subset $X\subseteq V$ and every set of pairs $P\subseteq V\times V$ the number of pairs $(x,(y,z))\in X\times P$ with $\{x,y,z\}$ being a hyperedge of $H$ is in the interval $d|X||P|\pm\eta|V|^3$. We show that for any $\varepsilon>0$ there exists $\eta>0$ such that every sufficiently large $(1/2+\varepsilon,\eta)$-quasirandom hypergraph contains a tetrahedron, i.e., four vertices spanning all four hyperedges. A known random construction shows that the density $1/2$ is best possible. This result is closely related to a question of Erd\H{o}s, whether every weakly quasirandom $3$-uniform hypergraph $H$ with density bigger than $1/2$, i.e., every large subset of vertices induces a hypergraph with density bigger than $1/2$, contains a tetrahedron.<br />Comment: 18 pages, second version addresses changes arising from the referee reports
- Subjects :
- Mathematics - Combinatorics
05C35 (primary), 05C65, 05C80 (secondary)
Subjects
Details
- Database :
- arXiv
- Journal :
- J. Combin. Theory Ser. B 121 (2016), 229-247
- Publication Type :
- Report
- Accession number :
- edsarx.1602.02289
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.jctb.2016.06.008