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Excluding a line from gammoids
Abstract
For all positive integers ℓ and r, we prove that a finite simple rank-r gammoid with no U2,ℓ+2-minor has at most ℓ(r−1)+1 elements. This gives an affirmative answer to Problem~7.2 of Boretsky and Walsh [European J. Combin.\ 137 (2026), 104419], for both gammoids and transversal matroids. The bound is sharp for every ℓ and r. The key estimate is that, for every finite simple gammoid M, the sum of |L|−2 over its long lines is at most $|E(M)|-r(M)$.
Related
BibTeX
@misc{chen2026excluding,
title = {{Excluding a line from gammoids}},
author = {Zhen Chen and Zhuo Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21315},
eprint = {2609.21315},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21315},
}
RIS
TY - UNPB
TI - Excluding a line from gammoids
AU - Zhen Chen
AU - Zhuo Li
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21315
UR - https://arxiv.org/abs/2609.21315
AB - For all positive integers ℓ and r, we prove that a finite simple rank-r gammoid with no U_2,ℓ+2-minor has at most ℓ(r−1)+1 elements. This gives an affirmative answer to Problem~7.2 of Boretsky and Walsh [European J. Combin.\ 137 (2026), 104419], for both gammoids and transversal matroids. The bound is sharp for every ℓ and r. The key estimate is that, for every finite simple gammoid M, the sum of |L|−2 over its long lines is at most $|E(M)|-r(M)$.
ER -