Ramsey Theory for Product Trees
Abstract
We develop a Ramsey theory for leaf-generated subsets of finite products of trees. Our starting point is a Theorem of Furstenberg and Weiss which states that for every k≥ 1 and α>0, if A is a subset of the leaves of Tn, where Tn is the complete binary tree of height n, with size |A|≥ 2αn, then for n sufficiently large the ancestor closed sub-tree TA⊂ Tn generated by A must contain a copy of Tk such that (1) all vertices in the same level of Tk are mapped into vertices at the same level of TA, (2) if a non-leaf vertex x∈ Tk is mapped into a vertex y ∈ TA, then the two children of x are mapped into descendants of the two children of y, and (3) the levels of TA occupied by the copy of Tk form an arithmetic progression. For the product Tn × Tn of two binary trees, we show that for every k≥ 1 and α> 1, for n sufficiently large, every subset A of the leaves of Tn× Tn of size at least 2αn has that its ancestor closure ΓA⊂ Tn × Tn contains similarly structured arithmetic copies of the height k four-ary tree, where the four children of every non-leaf vertex in the source tree are required to map below the four distinct \textit{diagonal children} of the image vertex, where a diagonal child of a vertex (x,y) ∈ Tn × Tn is obtained by moving one level down each in each component. Our results generalise to product of d-many finite b-ary trees with a sharp critical exponent of αcrit(d,b) =d−1 + logb(b−1). Geometrically, our results imply that any set E⊂ [0,1)d of upper Minkowski dimension greater than αcrit(d,b) contains arithmetic b-adic branching patterns of arbitrary finite order.
Related
BibTeX
@misc{fish2026ramsey,
title = {{Ramsey Theory for Product Trees}},
author = {Alexander Fish and Sean Skinner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28898},
eprint = {2609.28898},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28898},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ramsey Theory for Product Trees
AU - Alexander Fish
AU - Sean Skinner
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28898
UR - https://arxiv.org/abs/2609.28898
AB - We develop a Ramsey theory for leaf-generated subsets of finite products of trees. Our starting point is a Theorem of Furstenberg and Weiss which states that for every k≥ 1 and α>0, if A is a subset of the leaves of T_n, where T_n is the complete binary tree of height n, with size |A|≥ 2^αn, then for n sufficiently large the ancestor closed sub-tree T_A⊂ T_n generated by A must contain a copy of T_k such that (1) all vertices in the same level of T_k are mapped into vertices at the same level of T_A, (2) if a non-leaf vertex x∈ T_k is mapped into a vertex y ∈ T_A, then the two children of x are mapped into descendants of the two children of y, and (3) the levels of T_A occupied by the copy of T_k form an arithmetic progression. For the product T_n × T_n of two binary trees, we show that for every k≥ 1 and α> 1, for n sufficiently large, every subset A of the leaves of T_n× T_n of size at least 2^αn has that its ancestor closure Γ_A⊂ T_n × T_n contains similarly structured arithmetic copies of the height k four-ary tree, where the four children of every non-leaf vertex in the source tree are required to map below the four distinct \textit{diagonal children} of the image vertex, where a diagonal child of a vertex (x,y) ∈ T_n × T_n is obtained by moving one level down each in each component. Our results generalise to product of d-many finite b-ary trees with a sharp critical exponent of α_crit(d,b) =d−1 + log_b(b−1). Geometrically, our results imply that any set E⊂ [0,1)^d of upper Minkowski dimension greater than α_crit(d,b) contains arithmetic b-adic branching patterns of arbitrary finite order.
ER -