Independence number of hypergraphs under degree conditions

TL;DR

This paper extends known bounds on the size of independent sets in hypergraphs by weakening girth conditions and exploring degree constraints, providing new bounds and insights into hypergraph independence numbers.

Contribution

It introduces new bounds on independent set sizes in hypergraphs under weaker cycle and degree conditions, advancing understanding of hypergraph independence.

Findings

Independent sets of size proportional to n(log t)^{1/(k-1)}/t under weaker girth conditions.

Existence of large independent sets in hypergraphs with bounded (k-2)-degree.

New upper bounds for the (k-2)-degree Turán density of complete k-graphs.

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.