Hypergraph independence bounds: from maximum degree to average degree

TL;DR

This paper establishes a transfer theorem linking maximum degree and average degree bounds for the independence number in hereditary hypergraph classes, with applications to various graph families.

Contribution

It introduces a transfer theorem that converts maximum degree bounds into average degree bounds for independence numbers in hypergraphs.

Findings

Proves a transfer theorem for hereditary hypergraph classes.

Derives new bounds for independence numbers in specific graph families.

Connects degree-based bounds to coloring and fractional coloring results.

Abstract

We prove a transfer theorem for hereditary classes of $(r+1)$-uniform hypergraphs. Let $\mathcal H$ be such a class, and for $H\in\mathcal H$ write $\Delta(H)$ and $d(H)$ for the maximum degree and average degree of $H$, respectively. We show that, for every nearly logarithmic function $f$ in the sense defined below, a maximum-degree lower bound for the independence number of the form \[ \alpha(H)\ge (1-o(1))\frac{f(\Delta(H))}{\Delta(H)^{1/r}}|V(H)| \qquad\text{as }\Delta(H)\to\infty \] for all $H\in\mathcal H$ implies the corresponding average-degree lower bound \[ \alpha(H)\ge (1-o(1))\frac{f(d(H))}{d(H)^{1/r}}|V(H)| \qquad\text{as }d(H)\to\infty . \] We combine this transfer theorem with known coloring and fractional-coloring bounds to obtain consequences for graphs excluding a fixed cycle, graphs with bounded clique number, locally $q$-colorable graphs, and locally sparse uniform hypergraphs.