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New Lower Bound on Ramsey Number Shows Exponential Improvement

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10mo ago· 1 min readenNews

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Summary

The article presents a new lower bound on the Ramsey number for specific parameters, demonstrating an exponential improvement over previous bounds established by Erdős in 1947.

Key quotes

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There exists \$ \varepsilon=\varepsilon(C)> 0\$ such that \$ r(\ell, C\ell) \\geq \\left(p_C^{-1/2} + \varepsilon\right)^\ell\$.

This provides the first exponential improvement over the classical lower bound obtained by Erdős in 1947.

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We prove a new lower bound on the Ramsey number $r(\ell, C\ell)$ for any constant $C > 1$ and sufficiently large $\ell$, showing that there exists $\varepsilon=\varepsilon(C)> 0$ such that \[ r(\ell, C\ell) \geq \left(p_C^{-1/2} + \varepsilon\right)^\ell,

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