New lower bounds for the independence number of sparse graphs and hypergraphs

TL;DR

This paper establishes new lower bounds on the independence number of sparse graphs and hypergraphs, extending existing methods and answering longstanding questions in combinatorics.

Contribution

It introduces an extended proof technique for lower bounds on independence numbers, providing new results and a simplified proof of prior theorems.

Findings

New lower bounds for independence numbers in $K_r$-free graphs

Lower bounds for linear $k$-uniform hypergraphs

Simplified proof of Caro and Tuza's main result

Abstract

We obtain new lower bounds for the independence number of $K_r$-free graphs and linear $k$-uniform hypergraphs in terms of the degree sequence. This answers some old questions raised by Caro and Tuza \cite{CT91}. Our proof technique is an extension of a method of Caro and Wei \cite{CA79, WE79}, and we also give a new short proof of the main result of \cite{CT91} using this approach. As byproducts, we also obtain some non-trivial identities involving binomial coefficients.