Coordinate-extension degrees and layered k-uniform hypergraphs
Abstract
Let Palt=(C,T) be a k-palette. For 0≤ t≤ k−1, its tth coordinate-extension degree is the minimum, over every choice of t coordinates and every assignment of colors to them, of the proportion of assignments to the remaining k−t coordinates that complete the fixed colors to an admissible k-tuple. For a k-graph F, we define πtext(F) as the supremum of this degree over all palettes not admitted by F. We prove that πtext(F)=0 if and only if F is t−layered. We also relate t-layeredness to vanishing orders, min-layeredness, max-layeredness, and layeredness. These results recover and extend previous characterizations of Reiher, Rödl, and Schacht and of Lamaison, and answer a question of Lamaison for $3$-graphs. At t=0, the parameter π0ext(F) is the (k−2)-uniform Turán density πk−2(F). For every k≥3 and r≥2, we construct a finite k-graph Fk,r with πk−2(Fk,r)=2(r−1)/rkk. Thus 2/kk is an accumulation point for single forbidden k-graphs. We also show that the least density of a k-graph that fails condition Sp of Lin, Wang and Zhou is 4/(3kk). Finally, for every admissible matching of size m, we construct a k-graph that satisfies Sp for every coordinate pair, has no vanishing order, and has density 2m/kk. This disproves a conjecture of Lin, Wang and Zhou for every k≥3.
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BibTeX
@misc{yang2026coordinateextension,
title = {{Coordinate-extension degrees and layered $k$-uniform hypergraphs}},
author = {Jiabao Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03347},
eprint = {2610.03347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03347},
}
RIS
TY - UNPB
TI - Coordinate-extension degrees and layered k-uniform hypergraphs
AU - Jiabao Yang
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03347
UR - https://arxiv.org/abs/2610.03347
AB - Let Palt=(C,T) be a k-palette. For 0≤ t≤ k−1, its tth coordinate-extension degree is the minimum, over every choice of t coordinates and every assignment of colors to them, of the proportion of assignments to the remaining k−t coordinates that complete the fixed colors to an admissible k-tuple. For a k-graph F, we define π_t^ext(F) as the supremum of this degree over all palettes not admitted by F. We prove that π_t^ext(F)=0 if and only if F is t−layered. We also relate t-layeredness to vanishing orders, min-layeredness, max-layeredness, and layeredness. These results recover and extend previous characterizations of Reiher, Rödl, and Schacht and of Lamaison, and answer a question of Lamaison for $3$-graphs. At t=0, the parameter π_0^ext(F) is the (k−2)-uniform Turán density π_k−2(F). For every k≥3 and r≥2, we construct a finite k-graph F_k,r with π_k−2(F_k,r)=2(r−1)/rk^k. Thus 2/k^k is an accumulation point for single forbidden k-graphs. We also show that the least density of a k-graph that fails condition Sp of Lin, Wang and Zhou is 4/(3k^k). Finally, for every admissible matching of size m, we construct a k-graph that satisfies Sp for every coordinate pair, has no vanishing order, and has density 2^m/k^k. This disproves a conjecture of Lin, Wang and Zhou for every k≥3.
ER -