A polynomial improvement for the odd cycle-complete Ramsey numbers

TL;DR

This paper improves the lower bounds on odd cycle-complete Ramsey numbers, showing they grow faster than previously established for large k and fixed odd cycles.

Contribution

It provides a polynomial enhancement to the known bounds of cycle-complete Ramsey numbers for all fixed odd cycles greater than 7.

Findings

Established a new lower bound for r(C_ell, K_k) with a positive epsilon term.

Demonstrated the growth rate of these Ramsey numbers exceeds previous estimates.

Focused on fixed odd cycles with length greater than 7.

Abstract

We give a polynomial improvement to the cycle-complete Ramsey numbers \[ r(C_{\ell},K_k) \geq k^{1+1/(\ell- 2) + \varepsilon_{\ell} + o(1)}, \] for all fixed odd $\ell > 7$ with $k \rightarrow \infty$, for some $\varepsilon_{\ell} > 0$.