A sparse canonical Rado theorem
Abstract
Canonical Ramsey theory studies structured colour patterns in colourings with arbitrarily many colours. Lefmann's canonical Rado theorem extends the canonical van der Waerden theorem of Erdős and Graham to partition-regular linear systems. Recently, Alvarado, Kohayakawa, Morris, Mota and Ortega proved a sparse canonical van der Waerden theorem for random set [n]p. In this paper, we prove a sparse canonical Rado theorem. For every irredundant partition-regular matrix A of positive rank with k≥3 columns, and every prescribed partition witnessing the columns condition, if p≥ Cn−1/m(A), then with probability at least 1−e−cnp every colouring of [n]p contains Ω(pk nk−rank(A)) canonical solutions of Ax=0 with distinct coordinates. A matching 0-statement holds when the prescribed partition has one or at least three blocks; the Schur equation shows that it can fail for two-block patterns. For every prescribed partition, we also establish a robust threshold at the same scale under deletion of a fixed positive proportion of the random set.
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BibTeX
@misc{lou2026sparse,
title = {{A sparse canonical Rado theorem}},
author = {Zhenzhen Lou and Guangfu Wang and Man Wu and Zhifei Yan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37053},
eprint = {2610.37053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37053},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A sparse canonical Rado theorem
AU - Zhenzhen Lou
AU - Guangfu Wang
AU - Man Wu
AU - Zhifei Yan
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.37053
UR - https://arxiv.org/abs/2610.37053
AB - Canonical Ramsey theory studies structured colour patterns in colourings with arbitrarily many colours. Lefmann's canonical Rado theorem extends the canonical van der Waerden theorem of Erdős and Graham to partition-regular linear systems. Recently, Alvarado, Kohayakawa, Morris, Mota and Ortega proved a sparse canonical van der Waerden theorem for random set [n]_p. In this paper, we prove a sparse canonical Rado theorem. For every irredundant partition-regular matrix A of positive rank with k≥3 columns, and every prescribed partition witnessing the columns condition, if p≥ Cn^−1/m(A), then with probability at least 1−e^−cnp every colouring of [n]_p contains Ω(p^k n^k−rank(A)) canonical solutions of Ax=0 with distinct coordinates. A matching 0-statement holds when the prescribed partition has one or at least three blocks; the Schur equation shows that it can fail for two-block patterns. For every prescribed partition, we also establish a robust threshold at the same scale under deletion of a fixed positive proportion of the random set.
ER -