Back to Search
Start Over
Independence number of hypergraphs under degree conditions
- Publication Year :
- 2022
-
Abstract
- A well-known result of Ajtai et al. from 1982 states that every $k$-graph $H$ on $n$ vertices, with girth at least five, and average degree $t^{k-1}$ contains an independent set of size $c n (\log t)^{1/(k-1)}/t$ for some $c>0$. In this paper we show that an independent set of the same size can be found under weaker conditions allowing certain cycles of length 2, 3 and 4. Our work is motivated by a problem of Lo and Zhao, who asked for $k\ge 4$, how large of an independent set a $k$-graph $H$ on $n$ vertices necessarily has when its maximum $(k-2)$-degree $\Delta_{k-2}(H)\le dn$. (The corresponding problem with respect to $(k-1)$-degrees was solved by Kostochka, Mubayi, and Varstra\"ete [Random Structures & Algorithms 44, 224--239, 2014].) In this paper we show that every $k$-graph $H$ on $n$ vertices with $\Delta_{k-2}(H)\le dn$ contains an independent set of size $c (\frac nd \log\log \frac nd)^{1/(k-1)}$, and under additional conditions, an independent set of size $c (\frac nd \log \frac nd)^{1/(k-1)}$. The former assertion gives a new upper bound for the $(k-2)$-degree Tur\'an density of complete $k$-graphs.<br />Comment: To appear in Random Structures & Algorithms
- Subjects :
- Mathematics - Combinatorics
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2205.02877
- Document Type :
- Working Paper