An Efficiently Computable Lower Bound for the Independence Number of Hypergraphs

TL;DR

This paper addresses efficiently bounding the independence number of $k$-uniform hypergraphs by introducing a new lower bound $\ell(G)$ that depends only on $n$ and $m$ and is computable via a decreasing ratio of binomial terms. The main result proves a fundamental inequality $\binom{n}{\alpha(G)+1} \le m\binom{n}{\alpha(G)-k+1}$, from which $\ell(G)$ is derived as $\min\{i: f(i)\le m\}$ with $f(i)=\frac{\binom{n}{i+1}}{\binom{n}{i-k+1}}$, ensuring $\ell(G)\le \alpha(G)$; equality cases include complete and empty hypergraphs. For $k=2$ the bound reduces to a closed-form expression, and $\ell(G)=k-1$ iff $G$ is complete, while $\ell(G)=n$ iff $m=0$. The paper compares $\ell(G)$ to existing bounds: it matches or is weaker than Turán for graphs, but for $k\ge 3$ it can strictly improve on Turan-Spencer, Caro-Tuza, and Csaba-Plick-Shokoufandeh bounds in many instances, supported by explicit constructions and asymptotic families, making it a valuable addition to efficiently computable bounds on independence numbers in hypergraphs.

Abstract

We introduce a lower bound for the independence number of an arbitrary $k$-uniform hypergraph that only depends on the number of vertices and number of edges of the hypergraph.

Theorems & Definitions (21)

• Lemma 1
• proof
• Remark 2
• Theorem 3
• proof
• Remark 4
• Corollary 5
• proof
• Remark 6
• Proposition 7