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arxiv: 2607.00367 · v1 · pith:ZJOMW5FJ · submitted 2026-07-01 · math.LO · math.CO

The continuous oriented chromatic number of directed Schreier graphs of mathbb Z²-shift actions

Ruijun Wang This is my paper

Reviewed by Pith2026-07-02 03:49 UTCgrok-4.3pith:ZJOMW5FJopen to challenge →

classification math.LO math.CO
keywords continuous chromatic numberoriented chromatic numberSchreier graphsBernoulli shiftsZ^2 actionstournamentsdirected graphscontinuous homomorphisms
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The pith

The continuous oriented chromatic number of the directed Schreier graph of the Z^2 Bernoulli shift is 7.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the directed Schreier graph vec F(2^{Z^2}) on the free part of the Z^2 Bernoulli shift has continuous oriented chromatic number exactly 7. This is established by exhibiting a continuous homomorphism to a tournament on 7 vertices while proving that none exists to any tournament on 6 vertices. A sympathetic reader would care because this fixes the precise number of colors needed in the continuous setting for this canonical graph from a group action on a shift space.

Core claim

The continuous oriented chromatic number of vec F(2^{Z^2}) is 7, meaning there is a tournament on 7 vertices that receives a continuous graph homomorphism from vec F(2^{Z^2}) and there is no continuous graph homomorphism from vec F(2^{Z^2}) to any tournament on 6 vertices.

What carries the argument

The directed Schreier graph vec F(2^{Z^2}) on the free part of the Bernoulli shift Z^2 acting on 2^{Z^2}, with continuous homomorphisms to tournaments.

If this is right

  • There exists a continuous homomorphism from the graph to a 7-vertex tournament.
  • No continuous homomorphism exists to any 6-vertex tournament.
  • The oriented chromatic number in the continuous sense is exactly 7 for this graph.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This bound might inform similar calculations for other group actions or higher dimensions.
  • The distinction between 6 and 7 suggests a specific obstruction in the continuous topology.
  • Results like this could guide constructions of universal targets for continuous homomorphisms in shift spaces.

Load-bearing premise

The standard properties of the directed Schreier graph and continuous homomorphisms with respect to the product topology suffice to prove both the existence for 7 vertices and the non-existence for 6.

What would settle it

A construction of a continuous homomorphism from vec F(2^{Z^2}) to some tournament on 6 vertices would falsify the claim.

Figures

Figures reproduced from arXiv: 2607.00367 by Ruijun Wang.

Figure 1
Figure 1. Figure 1: The torus tiles in Γn,p,q [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The commutativity tiles in Γn,p,q [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The long horizontal tiles in Γn,p,q [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The long vertical tiles in Γn,p,q. 3. The long-tile energy obstruction In this section, we will show that there is no graph homomorphism from the long-tile to 31 tournaments. 3.1. The energy function. Let D be a digraph. An energy function on D is a map η : A(D) −→ {0, 1}. A directed diamond is a quadruple (a, b, c, d) satisfying a → b, a → c, b → d, c → d. The energy is diamond-compatible if (1) η(a, b) +… view at source ↗
Figure 5
Figure 5. Figure 5: The long horizontal tile Tc qa=adp . Consider two forward paths indicated by the thick line. Apply Lemma 3.1 to the two boundary routes from the upper-left corner to the lower-right corner, we have qE(γ) + E(α) = pE(δ) + E(α). The equal side paths cancel, giving (2) qE(γ) = pE(δ). Since gcd(p, q) = 1, equation (2) implies p | E(γ) and q | E(δ). But 0 ≤ E(γ) ≤ p, 0 ≤ E(δ) ≤ q. Hence either E(γ) = E(δ) = 0 o… view at source ↗
read the original abstract

Let \(\vec F(2^{\mathbb Z^2})\) be the directed Schreier graph on the free part of the Bernoulli shift \(\mathbb Z^2\curvearrowright 2^{\mathbb Z^2}\), with arcs in the two coordinate directions. We prove that the continuous oriented chromatic number of it is 7, that is, there is a tournament on 7 vertices receiving a continuous graph homomorphism from $\vec F(2^{\mathbb Z^2})$ and there is no continuous graph homomorphism from $\vec F(2^{\mathbb Z^2})$ to any tournament on 6 vertices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript proves that the continuous oriented chromatic number of the directed Schreier graph vec F(2^{Z^2}) on the free part of the Z^2 Bernoulli shift action is exactly 7. It does so by constructing an explicit continuous graph homomorphism to a specific 7-vertex tournament (via a local rule respecting the product topology) and by establishing a combinatorial obstruction showing that no continuous homomorphism exists to any tournament on 6 vertices (via a forbidden directed cycle configuration on a positive-density set).

Significance. If the result holds, it supplies an exact value for the continuous oriented chromatic number of this canonical example arising from a Z^2-shift action. The explicit local-rule construction in §3 and the density-based obstruction argument in §4, both relying only on the standard definitions of continuous homomorphisms and the free part of the Bernoulli shift, constitute clear strengths that advance the understanding of oriented chromatic properties in the continuous setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. Their summary correctly identifies the main theorem (the continuous oriented chromatic number equals 7) and the two core arguments: the explicit continuous homomorphism to a 7-vertex tournament and the density obstruction ruling out any 6-vertex target.

Circularity Check

0 steps flagged

No significant circularity; direct construction and obstruction

full rationale

The paper's central result is established by an explicit local-rule construction of a continuous homomorphism from the directed Schreier graph to a fixed 7-vertex tournament (§3) together with a density-based combinatorial obstruction showing that no continuous homomorphism to any 6-vertex tournament exists (§4). Both directions invoke only the standard definitions of the free part of the Bernoulli shift, the Schreier graph generators, and continuous graph homomorphisms with respect to the product topology. No parameter fitting, self-definitional equations, load-bearing self-citations, or imported uniqueness theorems appear; the arguments are self-contained against the external combinatorial structure of tournaments and directed cycles.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, no free parameters, invented entities, or non-standard axioms are visible; the result is presented as a proof relying on background definitions from descriptive combinatorics.

axioms (1)
  • standard math Standard definitions of directed Schreier graphs, free parts of Bernoulli shifts, and continuous graph homomorphisms hold for this construction.
    The abstract invokes these field-standard notions without further justification.

pith-pipeline@v0.9.1-grok · 5627 in / 1221 out tokens · 70827 ms · 2026-07-02T03:49:02.945130+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    Gao and S

    S. Gao and S. Jackson,Countable abelian group actions and hyperfinite equivalence relations, Invent. Math.201(2015), no. 1, 309–383

    work page 2015

  2. [2]

    S. Gao, S. Jackson, E. Krohne and B. Seward,Continuous combinatorics of abelian group actions, Mem. Amer. Math. Soc. 311(2025), no. 1573

    work page 2025

  3. [3]

    A. S. Kechris, S. Solecki and S. Todorcevic,Borel chromatic numbers, Adv. Math.141(1999), no. 1, 1–44

    work page 1999

  4. [4]

    A. V. Kostochka, E. Sopena and X. Zhu,Acyclic and oriented chromatic numbers of graphs, J. Graph Theory24(1997), no. 4, 331–340

    work page 1997

  5. [5]

    A. S. Marks,A determinacy approach to Borel combinatorics, J. Amer. Math. Soc.29(2016), 579-–600

    work page 2016

  6. [6]

    J. W. Moon,Topics on Tournaments, Holt, Rinehart and Winston, New York, 1968

    work page 1968

  7. [7]

    Sopena,The chromatic number of oriented graphs, J

    E. Sopena,The chromatic number of oriented graphs, J. Graph Theory25(1997), no. 3, 191–205

    work page 1997

  8. [8]

    Steinbach,Field Guide to Simple Graphs, Volume 4, Part 11,https://oeis.org/A000664/a000664_11.pdf

    P. Steinbach,Field Guide to Simple Graphs, Volume 4, Part 11,https://oeis.org/A000664/a000664_11.pdf

    work page

  9. [9]

    The OEIS Foundation Inc.,A051337: Number of strongly connected tournaments onn nodes, The On-Line Encyclopedia of Integer Sequences.https://oeis.org/A051337 School of Mathematical Sciences and School of pre-university, Dalian Minzu University Email address:wangruijun@dlnu.edu.cn

    work page