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Planarity and dimension I

Heather Smith Blake, Jędrzej Hodor, Piotr Micek, Michał T. Seweryn, William T. Trotter

TL;DR

This work proves that posets with planar cover graphs have dimension bounded polynomially in their standard example number, resolving a long-standing conjecture by showing dim(P) ≤ 64 se(P)^6 (se(P)+3)^2 + 12 with an accompanying polynomial-time embedding algorithm whose dimension is O(se(P)^8). The authors develop a multi-layered approach: unfold the poset, reduce to maximal good instances via a topology-informed framework of shadows and regions, and bound dimension through a suite of six auxiliary oriented graphs whose colorings decompose the problem into reversible pieces. A central technical achievement is the Coloring Lemma, which ensures that color classes formed from the auxiliary graphs yield reversible sets, thereby bounding dim(P). The results not only address the planar-cover-graph case but also align with broader themes of dim-boundedness for minor-closed graph classes and related structure theorems in poset theory, while offering a constructive embedding pathway amenable to algorithms. The paper lays the groundwork for Planarity and dimension II and III by indicating how large standard-examples and Kelly posets must appear inside high-dimensional planar-cover posets, and it situates the approach within a robust topological/combinatorial toolkit (regions, shadows, unfolded instances, and auxiliary digraph colorings).

Abstract

The dimension of a partially ordered set $P$ (poset for short) is the least positive integer $d$ such that $P$ is isomorphic to a subposet of $\mathbb{R}^d$ with the natural product order. Dimension is arguably the most widely studied measure of complexity for posets, and standard examples in posets are the canonical structure forcing dimension to be large. In many ways, dimension for posets is analogous to chromatic number for graphs with standard examples in posets playing the role of cliques in graphs. However, planar graphs have chromatic number at most four, while posets with planar diagrams may have arbitrarily large dimension. The key feature of all known constructions of such posets is that large dimension is forced by a large standard example. The question of whether every poset of large dimension and with a planar cover graph contains a large standard example has been a critical challenge in posets theory since the early 1980s, with very little progress over the years. We answer the question in the affirmative. Namely, we show that every poset $P$ with a planar cover graph has dimension $\mathcal{O}(s^8)$, where $s$ is the maximum order of a standard example in $P$.

Planarity and dimension I

TL;DR

This work proves that posets with planar cover graphs have dimension bounded polynomially in their standard example number, resolving a long-standing conjecture by showing dim(P) ≤ 64 se(P)^6 (se(P)+3)^2 + 12 with an accompanying polynomial-time embedding algorithm whose dimension is O(se(P)^8). The authors develop a multi-layered approach: unfold the poset, reduce to maximal good instances via a topology-informed framework of shadows and regions, and bound dimension through a suite of six auxiliary oriented graphs whose colorings decompose the problem into reversible pieces. A central technical achievement is the Coloring Lemma, which ensures that color classes formed from the auxiliary graphs yield reversible sets, thereby bounding dim(P). The results not only address the planar-cover-graph case but also align with broader themes of dim-boundedness for minor-closed graph classes and related structure theorems in poset theory, while offering a constructive embedding pathway amenable to algorithms. The paper lays the groundwork for Planarity and dimension II and III by indicating how large standard-examples and Kelly posets must appear inside high-dimensional planar-cover posets, and it situates the approach within a robust topological/combinatorial toolkit (regions, shadows, unfolded instances, and auxiliary digraph colorings).

Abstract

The dimension of a partially ordered set (poset for short) is the least positive integer such that is isomorphic to a subposet of with the natural product order. Dimension is arguably the most widely studied measure of complexity for posets, and standard examples in posets are the canonical structure forcing dimension to be large. In many ways, dimension for posets is analogous to chromatic number for graphs with standard examples in posets playing the role of cliques in graphs. However, planar graphs have chromatic number at most four, while posets with planar diagrams may have arbitrarily large dimension. The key feature of all known constructions of such posets is that large dimension is forced by a large standard example. The question of whether every poset of large dimension and with a planar cover graph contains a large standard example has been a critical challenge in posets theory since the early 1980s, with very little progress over the years. We answer the question in the affirmative. Namely, we show that every poset with a planar cover graph has dimension , where is the maximum order of a standard example in .
Paper Structure (31 sections, 78 theorems, 140 equations, 55 figures)

This paper contains 31 sections, 78 theorems, 140 equations, 55 figures.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Preliminaries
  4. Graphs
  5. Orders and posets
  6. Dimension of posets
  7. Topology and planarity
  8. Ordering edges in a plane graph
  9. Unfolding
  10. Topology of an instance
  11. Orientations and regions
  12. Leftmost and rightmost witnessing paths
  13. Blocks and shadows
  14. Ordering elements in B
  15. Reduction to a maximal good instance
  16. ...and 16 more sections

Key Result

theorem 1

For every poset $P$ with a planar cover graph, $\dim(P) \leqslant 64s^6(s+3)^2 + 12$ where $s = \mathop{\mathrm{se}}\nolimits(P)$.

Figures (55)

  • Figure 1: The standard example of order $6$.
  • Figure 2: Left: The wheel of order $6$: it has a planar cover graph (the order relation goes inwards) and it contains a subposet isomorphic to the Kelly poset of order $6$. Right: The Kelly poset of order $6$: it has a planar diagram and contains a subposet isomorphic to $S_6$.
  • Figure 3: A sample application of the notation for path concatenation.
  • Figure 4: Left: We have $e_0 \prec e_1 \prec e_2 \prec e_3 \prec e_4$ in the $u$-ordering as well as in the $(u,e_0)$-ordering. On the other hand, $e_4 \prec e_0 \prec e_1$ in the $u$-ordering, which is not the case in the $(u,e_0)$-ordering. Right: Let $U=uu_1u_2u_3u_4u_5u_6$ and let $V = uu_1u_2u_3v_4v_5$. Both paths start in $u$, and neither is a prefix of the other. We have, $u[U]u_3 = u[V]u_3$. Let $e' = u_2u_3$. Then, $u_3u_4 \prec u_3v_4$ in the $(u_3,e')$-ordering, hence, $U \prec V$ in the $(u,e)$-ordering. Moreover, the sets $\{u_4,u_5,u_6\}$ and $\{v_4,v_5\}$ are disjoint, thus, $U$ and $V$ are $u$-consistent.
  • Figure 5: The unfolding of a poset from $z_0$. The drawing is a diagram, that is, for two elements connected by an edge, the lower is less than the higher in the poset. In all the following figures, when a segment has no direction and it is not stated otherwise, the comparability is diagram-like (go upwards).
  • ...and 50 more figures

Theorems & Definitions (190)

  • Conjecture
  • theorem 1
  • Conjecture
  • proposition 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proof
  • proposition 4
  • ...and 180 more