Embedding rooted blow-ups of tree posets
Abstract
A tree poset T is a poset whose Hasse diagram is a tree. Bukh proved that if a family ℱ⊆ 2[n] contains (h(T)−1+ε)binomn⌊ n/2⌋ sets, then ℱ contains a weak copy of T, where h(T) is the height of T, the number of elements in a longest chain of T. Several strengthenings and generalizations of this result have been obtained. We prove the following robust variant. For a tree poset T and x∈ T, the b-blow-up T(x,b) rooted at x is the tree poset that we obtain from T by replacing every element u by bd new elements, where d is the distance d(x,u) in the Hasse diagram of T and an edge uv with v being closer to x is replaced by edges such that every new copy of v is joined to b new copies of u such that these new copies form pairwise disjoint sets for the copies of v. We prove that for any tree poset T, x∈ T, and ε>0 there exists δ such that if ℱ⊆ 2[n] contains (h(T)−1+ε)binomn⌊ n/2⌋ sets, then ℱ contains a weak copy of T(x,⌊ δn⌋). This settles a conjecture of Treglown and the author. As applications, we derive the known asymptotic counting and random versions of Bukh's theorem from this stronger embedding result, and obtain new maximal anti-Ramsey results for tree posets.
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BibTeX
@misc{patkos2026embedding,
title = {{Embedding rooted blow-ups of tree posets}},
author = {Balázs Patkós},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23543},
eprint = {2609.23543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23543},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Embedding rooted blow-ups of tree posets
AU - Balázs Patkós
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23543
UR - https://arxiv.org/abs/2609.23543
AB - A tree poset T is a poset whose Hasse diagram is a tree. Bukh proved that if a family ℱ⊆ 2^[n] contains (h(T)−1+ε)binomn⌊ n/2⌋ sets, then ℱ contains a weak copy of T, where h(T) is the height of T, the number of elements in a longest chain of T. Several strengthenings and generalizations of this result have been obtained. We prove the following robust variant. For a tree poset T and x∈ T, the b-blow-up T(x,b) rooted at x is the tree poset that we obtain from T by replacing every element u by b^d new elements, where d is the distance d(x,u) in the Hasse diagram of T and an edge uv with v being closer to x is replaced by edges such that every new copy of v is joined to b new copies of u such that these new copies form pairwise disjoint sets for the copies of v. We prove that for any tree poset T, x∈ T, and ε>0 there exists δ such that if ℱ⊆ 2^[n] contains (h(T)−1+ε)binomn⌊ n/2⌋ sets, then ℱ contains a weak copy of T(x,⌊ δn⌋). This settles a conjecture of Treglown and the author. As applications, we derive the known asymptotic counting and random versions of Bukh's theorem from this stronger embedding result, and obtain new maximal anti-Ramsey results for tree posets.
ER -