Bibliographic Information: Hou, J., Hu, C., Li, H., Liu, X., Yang, C., & Zhang, Y. (2024). Toward a density Corr'{a}di--Hajnal theorem for degenerate hypergraphs. arXiv preprint arXiv:2311.15172v2.
Research Objective: This paper aims to determine the maximum number of edges a degenerate hypergraph can have if it does not contain more than a specified number of disjoint copies of a given smaller hypergraph. This problem is closely related to the well-known Turán problem in extremal graph theory.
Methodology: The authors employ a combinatorial approach, utilizing techniques from extremal graph theory and probabilistic methods. They analyze the structure of extremal graphs (graphs achieving the maximum number of edges under the given constraints) and provide constructions for different ranges of parameters.
Key Findings: The paper presents several theorems establishing upper and lower bounds for the maximum number of edges in degenerate hypergraphs with restricted matching numbers. These bounds are shown to be asymptotically tight for certain classes of graphs and hypergraphs. The authors identify three distinct intervals for the "forbidden matching number" (the maximum allowed number of disjoint copies of the smaller hypergraph), each exhibiting different extremal constructions.
Main Conclusions: The results provide significant insights into the structure of extremal degenerate hypergraphs with bounded matching numbers. The authors establish a connection between the maximum number of edges and the Turán number of the forbidden hypergraph. The findings contribute to a deeper understanding of the interplay between density conditions and the existence of substructures in hypergraphs.
Significance: This research advances the field of extremal graph theory by extending classical results like the Corr'{a}di--Hajnal Theorem and the Erdős–Stone–Simonovits Theorem to the realm of degenerate hypergraphs. It provides a framework for studying density conditions that guarantee the existence of large matchings in hypergraphs.
Limitations and Future Research: The paper primarily focuses on specific classes of degenerate hypergraphs. Further research could explore the problem for a wider range of hypergraphs and investigate the sharpness of the obtained bounds for more general cases. Additionally, extending the results to a full density version of the Corr'{a}di--Hajnal Theorem for hypergraphs remains an open problem.
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by Jianfeng Hou... at arxiv.org 10-15-2024
https://arxiv.org/pdf/2311.15172.pdfDeeper Inquiries