Infinitely many hypergraphs outperforming the Caro–Tuza bound

Prove that for every integer k 3, there exist infinitely many k-uniform hypergraphs G for which the efficiently computable lower bound (G) exceeds the Caro–Tuza lower bound CT(G) for the independence number.

Background

The paper introduces a lower bound (G) for the independence number of a k-uniform hypergraph that depends only on the number of vertices and edges. The Caro–Tuza lower bound CT(G), by contrast, uses the degrees of the individual vertices. The authors give examples of regular 3-uniform and 4-uniform hypergraphs in which (G) is strictly larger than CT(G).

Based on these examples and experimental evidence, the authors formulate the unresolved claim that this strict improvement occurs for infinitely many k-uniform hypergraphs for every k at least 3. The conjecture concerns existence and does not specify a construction or a particular infinite family.