Branch-Sensitive Density Increments for k-Configurations
Abstract
We refine the density-increment argument for k-configurations developed by Beker. The point is not to introduce a new inverse theorem, but to retain information that is lost when the graph-counting output is compressed into a single uniform alternative. For the transitive Kk appearing in the k-configuration problem we keep a fixed moment throughout the graph recursion and obtain three genuinely different outputs: a row-degree anomaly, a non-centred grid anomaly, and a centred grid anomaly. The centred case is treated directly through positivity of the associated Gram moments, avoiding the usual unbalancing step. Each output is then routed through the appropriate local Kelley--Meka mechanism with its own density gain, rank cost, and width cost. A weighted iteration budget prevents the most expensive local branches from being charged the maximal number of times. For a finite abelian group G of odd order and A⊆ G of density α, write L=log(2/α) and Λ=L+log(2k). We obtain a lower bound of the form Ppx1,…,xk∈ G ((xi+xj)/2∈ A for all 1≤ i≤ j≤ k) ≥ exp (−CΦ(k,L,Λ)), where Φ(k,L,Λ)=k36L6Λ6+k34L8Λ5+k32L14Λ+k30L16. Consequently, in the Erdős--Moser application one may replace the exponent $68$ in Beker's k-configuration route by $36$; in particular the same reduction yields the sum-free lower bound with every exponent c<1/36.
Related
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BibTeX
@misc{zhi2026branchsensitive,
title = {{Branch-Sensitive Density Increments for $k$-Configurations}},
author = {Yao Zhi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03750},
eprint = {2610.03750},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03750},
}
RIS
TY - UNPB
TI - Branch-Sensitive Density Increments for k-Configurations
AU - Yao Zhi
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.03750
UR - https://arxiv.org/abs/2610.03750
AB - We refine the density-increment argument for k-configurations developed by Beker. The point is not to introduce a new inverse theorem, but to retain information that is lost when the graph-counting output is compressed into a single uniform alternative. For the transitive K_k appearing in the k-configuration problem we keep a fixed moment throughout the graph recursion and obtain three genuinely different outputs: a row-degree anomaly, a non-centred grid anomaly, and a centred grid anomaly. The centred case is treated directly through positivity of the associated Gram moments, avoiding the usual unbalancing step. Each output is then routed through the appropriate local Kelley--Meka mechanism with its own density gain, rank cost, and width cost. A weighted iteration budget prevents the most expensive local branches from being charged the maximal number of times. For a finite abelian group G of odd order and A⊆ G of density α, write L=log(2/α) and Λ=L+log(2k). We obtain a lower bound of the form Pp_x_1,…,x_k∈ G ((x_i+x_j)/2∈ A for all 1≤ i≤ j≤ k) ≥ exp (−CΦ(k,L,Λ)), where Φ(k,L,Λ)=k^36L^6Λ^6+k^34L^8Λ^5+k^32L^14Λ+k^30L^16. Consequently, in the Erdős--Moser application one may replace the exponent $68$ in Beker's k-configuration route by $36$; in particular the same reduction yields the sum-free lower bound with every exponent c<1/36.
ER -