Bibliographic Information: Bennett, P., Delcourt, M., Li, L., & Postle, L. (2024). On generalized Ramsey numbers in the non-integral regime. arXiv preprint arXiv:2212.10542v3.
Research Objective: This research paper aims to improve the upper bound on generalized Ramsey numbers, specifically focusing on the non-integral regime where the exponent in the bound is non-integral.
Methodology: The authors utilize the Forbidden Submatching Method, a recently developed technique for finding perfect matchings in hypergraphs while avoiding specific submatchings. They introduce a novel potential function on submatchings to define the appropriate set of forbidden submatchings, enabling them to apply the method effectively.
Key Findings: The paper demonstrates that for fixed positive integers p and q, where p-2 is not divisible by (p choose 2)-q+1, the generalized Ramsey number f(n, p, q) has an improved upper bound of O((n^(p-2)/log n)^(1/((p choose 2)-q+1))). This result holds for a wide range of generalizations, including replacing cliques with arbitrary graphs, extending to list coloring, and applying to k-uniform hypergraphs.
Main Conclusions: The authors successfully improve the general upper bound for generalized Ramsey numbers in the non-integral regime, surpassing the previously known bound obtained through the Lovász Local Lemma. This improvement is significant as it demonstrates that the local lemma bound is not tight in this regime.
Significance: This research significantly contributes to the field of Ramsey theory by providing a tighter upper bound for generalized Ramsey numbers in the non-integral regime. This result has implications for other areas of combinatorics, such as the problem of Brown, Erdős, and Sós concerning the minimum number of edges in hypergraphs with specific properties.
Limitations and Future Research: The authors acknowledge that their proof does not directly translate to the sparse regime where the maximum degree of the graph is significantly smaller than the number of vertices. Further research could explore tighter bounds in this regime. Additionally, investigating the potential difference in order of magnitude between generalized Ramsey numbers for coloring and list coloring presents an interesting avenue for future work.
Till ett annat språk
från källinnehåll
arxiv.org
Djupare frågor