Hypergraph independence bounds: from maximum degree to average degree

Abstract: We prove a transfer theorem for hereditary classes of $(r+1)$-uniform hypergraphs. Let $\mathcal G$ be such a class, and for $H\in\mathcal G$ write $Δ(H)$ 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 [ α(H)\ge (1-o(1))\frac{f(Δ(H))}{Δ(H)^{1/r}}|V(H)| \qquad\text{as }Δ(H)\to\infty ] for all $H\in\mathcal G$ implies the corresponding average-degree lower bound [ α(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.

• The paper introduces a transfer theorem that converts maximum-degree independence bounds into average-degree bounds for hereditary hypergraph classes.
• It employs an iterative cleaning process to remove high-degree vertices, achieving near-optimal independence numbers in sparse settings.
• The results extend to locally sparse graphs and weighted cases, unifying techniques from extremal combinatorics and probabilistic graph theory.

Hypergraph Independence Bounds: From Maximum Degree to Average Degree

Overview

This paper presents a general transfer theorem relating maximum-degree-based and average-degree-based lower bounds for independence numbers in hereditary classes of $(r+1)$-uniform hypergraphs. The transfer principle formalizes a widely used but previously ad hoc idea for deriving average-degree bounds from more tractable maximum-degree results, thereby systematizing a technique central to extremal combinatorics and probabilistic graph theory. The transfer is shown to be broadly applicable, yielding average-degree independence bounds that match known best maximum-degree analogues up to subpolynomial factors for a variety of sparse graph and hypergraph families, including $C_k$-free graphs, $K_{k+1}$-free graphs, locally colorable graphs, and locally sparse hypergraphs.

Main Theoretical Contribution

Let $H$ be an $(r+1)$-uniform hypergraph. Prior to this work, the typical lower bound on the independence number $\alpha(H)$ using a random deletion argument is

$$\alpha(H) = \Omega\left(\frac{|V(H)|}{d(H)^{1/r}}\right)$$

where $d(H)$ is the average degree. However, in many classes of sparse or locally constrained hypergraphs and graphs, improvements by logarithmic or polylogarithmic factors are achieved using local structure to enhance the random deletion or coloring analysis in the maximum degree regime.

A significant obstacle in leveraging maximum degree bounds in the context of average degree is that high-degree vertices undetectable via the average degree can invalidate the "maximum-degree-based" proofs. The main result establishes a formal transfer mechanism under mild regularity conditions on the class $\mathcal{G}$ (namely, hereditariness) and the bounding function $f$ (which must be "nearly logarithmic").

Main Theorem (paraphrased):

Suppose that for all $C_k$0 in a hereditary class $C_k$1 of $C_k$2-uniform hypergraphs,

$C_k$3

where $C_k$4 is "nearly logarithmic." Then,

$C_k$5

A function $C_k$6 is nearly logarithmic if (i) $C_k$7 as $C_k$8 and $C_k$9, and (ii) for all $K_{k+1}$0 and $K_{k+1}$1, $K_{k+1}$2 is stable up to small multiplicative perturbations in its argument over suitable ranges. In practice, functions such as $K_{k+1}$3, $K_{k+1}$4, $K_{k+1}$5, and $K_{k+1}$6 for $K_{k+1}$7 all qualify.

Proof Architecture

The proof employs a cleaning process iteratively deleting vertices of unusually high degree relative to the current average degree: at each step, a vertex with degree exceeding $K_{k+1}$8 times the current average degree is removed, steadily lowering the average degree. The key arguments ensure that either the average degree drops quickly—making the trivial random bound effective—or so many vertices are cleaned that the denominator shrinks subpolynomially, or the induced subhypergraph’s maximum and average degrees are aligned, allowing the maximum-degree bound to be invoked.

Quantitative tracking of the functional growth of $K_{k+1}$9 during cleaning, coupled with the slow-variation property of "nearly logarithmic" $H$0, ensures that the parameters in the average-degree assertion can be driven arbitrarily close to their maximum-degree analogues.

Applications

Graphs Excluding Cycles or Cliques

By combining the main theorem with sharp maximum-degree chromatic bounds due to Davies, Kang, Pirot, and Sereni as well as the fundamental Ajtai-Komlós-Szemerédi and Shearer independence theorems, the following results are obtained:

• For every fixed $H$1, any $H$2-free graph $H$3 of average degree $H$4 satisfies:

$H$5

• For $H$6-free graphs, improved lower bounds involving additional $H$7 factors and dependencies on clique number.

Locally Colorable and Locally Sparse Hypergraphs

Using recent entropy-based fractional coloring bounds, the following is established:

• For locally $H$8-colorable graphs of average degree $H$9:

$(r+1)$0

• For locally sparse $(r+1)$1-uniform hypergraphs:

$(r+1)$2

where $(r+1)$3 is an explicit constant.

Weighted Extensions

The cleaning argument generalizes directly to weighted independence numbers and appropriately defined weighted (maximum and average) degrees, allowing weighted analogues of all main results.

Implications and Future Directions

The transfer theorem is a robust method for reducing average-degree independence problems in hereditary hypergraph families to their maximum-degree analogues. Practically, this obviates the need for bespoke average-degree arguments in many sparse graph and hypergraph settings; one can systematically derive optimal or nearly optimal results via the transfer.

Theoretically, the result can be viewed as a unification and general principle underlying prior scattered uses of cleaning/deletion arguments, fractional chromatic methods, and independence bound transfers. It enables the extension of sophisticated probabilistic and entropy-based local techniques to regimes controlled previously only by average-degree heuristics.

Future work could explore:

• Generalization to broader structural classes, including non-uniform hypergraphs or non-hereditary properties.
• Tighter characterizations of the nearly logarithmic class, possibly extracting more explicit constants or identifying optimal functions.
• Algorithmic instantiations of the cleaning/deletion process for constructive and randomized algorithms achieving large independent sets in sparse combinatorial structures.

Conclusion

This paper provides a rigorous and broadly applicable transfer principle for independence lower bounds in hypergraphs, systematically enabling average-degree guarantees based on maximum-degree independence results for hereditary classes and nearly logarithmic comparison functions. The framework both clarifies existing techniques and unlocks a wide spectrum of improved combinatorial bounds relevant to extremal and probabilistic graph theory applications (2604.28046).