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arXiv · 2609.14525

Retained-Set Descent for Diagonal Ramsey Numbers

Zhipeng Lu·Sichen Wang
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Abstract

We study how far a fixed Ramsey upper bound can be improved by descending through blue neighborhoods in one vertex set while keeping a second set fixed. A weighted inequality in the two set sizes determines when the descent can stop. For the source bound specified here, the infimum diagonal exponent over all finite derivations lies in $[1.305,\,1.307]$. A finite derivation gives $R(k,k)\le3.69507^k$ for all sufficiently large $k$; a concave polygon proves the lower bound for every finite depth. We also characterize the infimum as a greatest fixed point and show that every larger exponent has a finite derivation valid uniformly for nearby clique-size ratios.

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Zhipeng Lu, Sichen Wang. 2026-09-13. Retained-Set Descent for Diagonal Ramsey Numbers. https://arxiv.org/abs/2609.14525

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