The Ramsey number of a long even cycle versus a star

TL;DR

This paper determines the exact Ramsey number for a long even cycle versus a star when parameters are large, linking it to Turán numbers and advancing understanding of extremal graph theory.

Contribution

It provides the exact value of $R(C_{2ell},K_{1,n})$ for large parameters, connecting Ramsey and Turán numbers in new asymptotic regimes.

Findings

Exact Ramsey number for long even cycle versus star

Connection between Ramsey and Turán numbers for large cycles

Advancement in extremal graph theory understanding

Abstract

We find the exact value of the Ramsey number $R(C_{2\ell},K_{1,n})$, when $\ell$ and $n=O(\ell^{10/9})$ are large. Our result is closely related to the behaviour of Tur\'an number $ex(N, C_{2\ell})$ for an even cycle whose length grows quickly with $N$.