Turán Number of Disjoint Berge Paths
Analysis
This paper investigates the Turán number for Berge paths in hypergraphs. Specifically, it determines the exact value of the Turán number for disjoint Berge paths under certain conditions on the parameters (number of vertices, uniformity, and path length). This is a contribution to extremal hypergraph theory, a field concerned with finding the maximum size of a hypergraph avoiding a specific forbidden subhypergraph. The results are significant for understanding the structure of hypergraphs and have implications for related problems in combinatorics.
Key Takeaways
- •Determines the exact Turán number for disjoint Berge paths under specific conditions.
- •Contributes to the field of extremal hypergraph theory.
- •Provides insights into the structure of hypergraphs.
“The paper determines the exact value of $\mathrm{ex}_r(n, ext{Berge-} kP_{\ell})$ when $n$ is large enough for $k\geq 2$, $r\ge 3$, $\ell'\geq r$ and $2\ell'\geq r+7$, where $\ell'=\left\lfloor rac{\ell+1}{2} ight floor$.”