PreprintCC BY 4.0arXiv
Retained-Set Descent for Diagonal Ramsey Numbers
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)≤3.69507k 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.
Related
BibTeX
@misc{lu2026retainedset,
title = {{Retained-Set Descent for Diagonal Ramsey Numbers}},
author = {Zhipeng Lu and Sichen Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14525},
eprint = {2609.14525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14525},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Retained-Set Descent for Diagonal Ramsey Numbers
AU - Zhipeng Lu
AU - Sichen Wang
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14525
UR - https://arxiv.org/abs/2609.14525
AB - 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)≤3.69507^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.
ER -