Ramsey Obstructions to Disambiguation
Abstract
A partial matrix has entries in {0,1,⋆}, and a disambiguation replaces each ⋆ by $0$ or $1$. We construct partial matrices whose fully specified submatrices satisfy strong restrictions, yet every disambiguation contains every binary matrix of a prescribed size. Our first result answers a question of Alon, Hanneke, Holzman and Moran on the disambiguation of linear classifiers with margin. For 0<ε<π/2, let Mεd be the partial matrix indexed by points of the unit sphere 𝕊d, with entry $0$ for pairs at spherical distance at most ε, $1$ for pairs at distance at least π−ε, and ⋆ otherwise. Although these matrices have VC-dimension bounded independently of d, we prove that every disambiguation contains every binary k× k matrix once d is sufficiently large. This also yields a partial concept class of Littlestone dimension $1$ with no disambiguation of finite VC-dimension. We also construct, for every k, a finite partial matrix whose fully specified 2×2 submatrices are all constant, while every disambiguation contains every binary k× k matrix. A symmetric analogue holds for partial graphs: for every k, there exists a partial graph of VC-dimension at most $1$ whose fully specified induced subgraphs are all cliques or stable sets, yet every disambiguation contains every k-vertex graph as an induced subgraph. A disambiguation can be viewed as a $2$-coloring of the unspecified entries, making Ramsey theory a natural framework for forcing prescribed patterns. Our proofs draw on two recent Ramsey theorems: the geometric argument uses Pálvölgyi's Dense Block theorem, while the combinatorial constructions rely on the girth Ramsey theorem of Reiher and Rödl, a suitable strengthening of the induced Ramsey theorem.
Related
BibTeX
@misc{bourneuf2026ramsey,
title = {{Ramsey Obstructions to Disambiguation}},
author = {Romain Bourneuf and Antonin Kiladjian and Stéphan Thomassé},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16359},
eprint = {2609.16359},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16359},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ramsey Obstructions to Disambiguation
AU - Romain Bourneuf
AU - Antonin Kiladjian
AU - Stéphan Thomassé
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16359
UR - https://arxiv.org/abs/2609.16359
AB - A partial matrix has entries in {0,1,⋆}, and a disambiguation replaces each ⋆ by $0$ or $1$. We construct partial matrices whose fully specified submatrices satisfy strong restrictions, yet every disambiguation contains every binary matrix of a prescribed size. Our first result answers a question of Alon, Hanneke, Holzman and Moran on the disambiguation of linear classifiers with margin. For 0<ε<π/2, let M_ε^d be the partial matrix indexed by points of the unit sphere 𝕊^d, with entry $0$ for pairs at spherical distance at most ε, $1$ for pairs at distance at least π−ε, and ⋆ otherwise. Although these matrices have VC-dimension bounded independently of d, we prove that every disambiguation contains every binary k× k matrix once d is sufficiently large. This also yields a partial concept class of Littlestone dimension $1$ with no disambiguation of finite VC-dimension. We also construct, for every k, a finite partial matrix whose fully specified 2×2 submatrices are all constant, while every disambiguation contains every binary k× k matrix. A symmetric analogue holds for partial graphs: for every k, there exists a partial graph of VC-dimension at most $1$ whose fully specified induced subgraphs are all cliques or stable sets, yet every disambiguation contains every k-vertex graph as an induced subgraph. A disambiguation can be viewed as a $2$-coloring of the unspecified entries, making Ramsey theory a natural framework for forcing prescribed patterns. Our proofs draw on two recent Ramsey theorems: the geometric argument uses Pálvölgyi's Dense Block theorem, while the combinatorial constructions rely on the girth Ramsey theorem of Reiher and Rödl, a suitable strengthening of the induced Ramsey theorem.
ER -