A polynomial improvement for the odd cycle-complete Ramsey numbers
Marcelo Campos, Matthew Jenssen, Marcus Michelen, Florian Pfender, Julian Sahasrabudhe
TL;DR
This work establishes a polynomial improvement for the odd cycle–complete Ramsey numbers: for fixed odd $\ell>7$, $r(C_{\ell},K_k) \ge k^{1+1/(\ell-2)+\varepsilon_{\ell}+o(1)}$ with a positive $\varepsilon_{\ell}$ (explicitly $\varepsilon_{\ell}=((\ell-2)(2\ell-5))^{-1}$ in the main theorem). The authors construct a $C_{\ell}$-free graph with large order and small independence number by superimposing blow-ups of random graphs and applying a two-stage cleanup: a vertex-deletion step to remove non-simple broken cycles, followed by an apex-based edge-deletion step to eliminate remaining $C_{\ell}$ cycles. A robust pseudo-random framework is established via a high-probability event $\mathcal{A}$, and a spectral analysis bound on $J$–$J$ walks is used to control cycle counts and independence. The combination yields a lower bound beyond the deletion-threshold barrier and demonstrates a polynomial path to improving certain Ramsey-type exponents, with implications for similar extremal problems in random-graph constructions.
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 $\varepsilon_{\ell} > 0$.