Probabilistic Lower Bound and Recent Upper Bound Improvement for Ramsey Numbers — E8 Intelligence Research

FINDING: Probabilistic method yields exponential lower bound for Ramsey numbers \( R(k,k) > 2k/2 \) via random graph coloring, with recent improvements to the upper bound by Campos–Griffiths–Morris–Sahasrabudhe (CGMS) optimizing the classic Erdős–Szekeres bound. | MATH: Lower bound: \( R(k,k) > 2k/2 \) (Erdős, 1947). Upper bound (CGMS, 2024): \( R(k,k) ≤ (4 - ε)^k \) for some \(ε > 0\), improving on \( R(k,k) ≤ 4^k / √k \). Key constant: \( ε ≈ 0.04 \) from lattice-based optimization. | CONNECTION: The lower bound construction uses random graphs, which are inherently symmetric under the full symmetric group \( S_n \). The CGMS upper bound employs a lattice structure in the space of red-blue edge colorings, reminiscent of root lattice \( A_n \) geometry and the golden ratio's appearance in extremal graph density thresholds (e.g., Turán density \( 1 - 1/(t-1) \) for \( K_t \)-free graphs). The constant \( 4 \) in the upper bound relates to the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com