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The continuous oriented chromatic number of the directed Z^{2} Schreier graph is exactly 7; its Borel version for higher-rank shifts is 5.

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2026-07-14 16:51 UTC pith:ZJOMW5FJ

load-bearing objection Exact continuous oriented chromatic number 7 (and Borel 5) for directed Z^{2} Schreier graphs, via clean energy/marker casework and toast heights. the 2 major comments →

arxiv 2607.00367 v2 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
classification math.LO math.CO MSC 03E1505C1505C20
keywords continuous combinatoricsdirected Schreier graphoriented chromatic numberBernoulli shifttournamentsTwelve Tiles TheoremBorel toast
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper settles two oriented-coloring numbers for the free parts of Bernoulli shifts of abelian groups. For the directed Schreier graph of the Z^{2}-shift, continuous oriented colorings require seven colors: a concrete tournament on seven vertices receives a continuous homomorphism, while every tournament on six vertices is ruled out. For directed Schreier graphs of Z^{n}-shifts with n>1 the Borel (and measurable) oriented chromatic number is five. The work converts continuous questions into finite combinatorial checks via rectangular tiles, then obstructs six-vertex tournaments by energy identities on long tiles and marker propagation on tori, and builds the five- and seven-color maps from toast partitions and an explicit finite coloring.

Core claim

The continuous oriented chromatic number of the directed Schreier graph on the free part of the Bernoulli Z^{2}-shift is exactly 7, and the Borel oriented chromatic number of the corresponding directed Schreier graphs for Z^{n} with n>1 is exactly 5. In particular there is a continuous homomorphism into a fixed 7-vertex tournament, none into any 6-vertex tournament, a Borel homomorphism into the regular 5-vertex tournament, and none into the unique strong 4-vertex tournament.

What carries the argument

The Directed Twelve Tiles Theorem reduces continuous oriented colorings of the infinite graph to ordinary graph homomorphisms out of twelve finite rectangular tiles. Long-tile energy functions (order energies plus four non-order energies) force monochromatic cycles of coprime lengths, while torus-tile marker sets for directed length-2 paths are forced to be invariant under a transitive translation and therefore empty or full, both impossible.

Load-bearing premise

The argument treats the directed twelve-tiles equivalence as exact for oriented targets: continuous homomorphisms from the infinite graph exist precisely when the finite tile graphs map into the target tournament.

What would settle it

Exhibit either a continuous homomorphism from the free Z^{2}-shift graph into some six-vertex tournament, or a continuous homomorphism into a five-vertex tournament; either would contradict the claimed continuous number 7. Alternatively, produce a Borel homomorphism into a four-vertex tournament for some n>1.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper determines the continuous oriented chromatic number of the directed Schreier graph of the free part of the Bernoulli Z^{2}-shift: χ_co(⃗F(2^{Z^{2}})) = 7. The upper bound is an explicit continuous homomorphism to a concrete 7-vertex tournament R₇, obtained by exhibiting a homomorphism ⃗Γ_{1,3,4} → R₇ and invoking the Directed Twelve Tiles Theorem. The lower bound shows there is no continuous homomorphism to any tournament on 6 vertices by reducing, via the same theorem, to finite checks on two of the twelve tiles against the classical list of 35 strong 6-tournaments: an admissible energy (order energy for 27 tournaments, four non-order energies for four more) obstructs the long tile, while a marker-shift/row-nontriviality argument on directed length-2 paths obstructs the torus tile for the remaining four. Separately, the paper proves that the Borel (and measurable) oriented chromatic number of ⃗F(2^{Z^n}) for n > 1 is 5, with lower bound by an ergodicity argument against the unique strong 4-tournament and upper bound by gluing model height functions along a Borel toast.

Significance. If correct, the continuous result is a sharp determination of a natural continuous combinatorial invariant for the directed Z^{2}-Schreier graph, sitting cleanly between the known continuous chromatic number 4 of the undirected graph and the classical oriented chromatic numbers of grids. The Borel result χ_Bo = 5 for all n > 1 is likewise sharp and uses the toast technology in a transparent way. Strengths include the exhaustive finite classification against the 35 strong 6-tournaments, the explicit certificates (interval criterion tables, energy tables, de Bruijn level sets, and the full 7-coloring of Γ_{1,3,4} in the appendices), and the clean separation of continuous and Borel regimes. The work is a natural and substantial contribution to continuous and Borel combinatorics of abelian group actions.

major comments (2)
  1. §4.1 Lemma 4.1 (Marker shift) and §4.2 Lemma 4.2 (Row nontriviality): the local transfer law and the acyclicity of the two de Bruijn digraphs D_ε(T) for T ∈ {T_A, T_B} are asserted after listing transfer classes and level sets, but the verification is described only as “tedious like playing a Sudoku” and is not machine-checked or fully expanded. These two lemmas are load-bearing for the four remaining tournaments (and their reversals) that escape the energy method; without a complete check that every 2×4 rectangle preserves the claimed classes and that every de Bruijn arc strictly increases level, the torus obstruction (Prop. 4.3) and therefore the continuous lower bound χ_co > 6 are not fully established. A short computer certificate or an expanded case table would close the gap.
  2. §3.2–3.3 and Appendix A: the 27 interval-criterion certificates and the four non-order energy certificates are presented as exhaustive against the classical list of 35 strong 6-tournaments. While the codes and out-neighborhoods are listed, the paper does not record an independent verification that the 35 codes are complete and pairwise non-isomorphic under the canonical-code definition of §2.2. A one-line reference to a standard enumeration (or a short script confirming the codes) would remove residual doubt that some strong tournament was missed or misclassified.
minor comments (5)
  1. Abstract and Theorem 1.2: the abstract claims both Borel and measurable oriented chromatic number equal 5; the body proves Borel = 5 and notes the lower bound is in fact measurable, but never states a separate measurable upper bound. A sentence clarifying that the toast construction is Borel (hence measurable) would align the abstract with the text.
  2. §2.2: the canonical-code definition uses a non-standard enumeration of pairs p_k; a brief remark that the resulting codes match a known OEIS or Moon enumeration would help the reader trust the list of 35.
  3. §5 and Appendix B: the 7-coloring of Γ_{1,3,4} is given as twelve rectangular arrays; a single sentence confirming that all horizontal and vertical adjacencies were checked against the arc set of R₇ would make the certificate self-contained.
  4. §6.2: the toast separation is fixed at R = 100; the extension criterion only needs distance ≥ 14 or so. A remark that any sufficiently large R works would avoid the impression that 100 is special.
  5. Typographical: “T welve Tiles Theorem” (p. 4), “chroma tic” in the title, and occasional missing spaces after commas in the tournament tables should be cleaned.

Circularity Check

0 steps flagged

No significant circularity: continuous and Borel bounds rest on external black-box theorems plus independent finite combinatorial certificates and an explicit height-function construction.

full rationale

The continuous claim χ_co(vec F(2^{Z^{2}}))=7 is obtained by (i) an explicit finite homomorphism Γ̅_{1,3,4} o R_7 that lifts via the Directed Twelve Tiles Theorem (Thm 2.1, specialization of Gao–Jackson–Krohne–Seward) and (ii) exhaustive obstruction of all 35 strong 6-tournaments by admissible energies on the long tile (Prop. 3.2, Cor. 3.6) and marker-shift/row-nontriviality on the torus tile (Lems 4.1–4.2, Prop. 4.3). Both obstruction families are local combinatorial statements whose out-neighborhoods, energy tables, de Bruijn level sets and interval certificates are fully listed in the paper (Tables 2–29, App. A–B); they do not presuppose the continuous chromatic number. The Borel claim χ_Bo=5 uses the external Borel toast theorem of Gao–Jackson–Krohne–Seward together with an independent extension criterion (Lem. 6.3) and phase-lift (Lem. 6.4) for height functions; the lower bound is a short ergodicity argument on the unique strong 4-tournament. Self-citations appear only as background tools already established for digraphs; none of them is load-bearing for the target numerical values. The derivation is therefore self-contained against its stated external inputs and exhibits no definitional, fitted-input, or uniqueness-import circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 2 invented entities

The paper works inside standard descriptive set theory and finite tournament theory. Load-bearing external inputs are the Directed Twelve Tiles Theorem, the Borel toast theorem, the classical enumeration of 35 strong 6-tournaments, and elementary ergodicity. No numerical free parameters are fitted; the separation constant R=100 is an arbitrary large enough integer. Invented technical devices (admissible energy, marker sets M_T) are defined and verified inside the paper.

free parameters (1)
  • toast separation R = 100
    Chosen as R=100 (or any sufficiently large fixed integer) so that ℓ1-distance ≥ 100 guarantees the profile-extension inequality in Lemma 6.5; not fitted to data.
axioms (4)
  • domain assumption Directed Twelve Tiles Theorem: continuous homomorphisms vec F(2^{Z^{2}}) o H exist iff finite homomorphisms vec Γ_{n,p,q} o H exist for coprime p,q (Theorem 2.1).
    Taken from Gao–Jackson–Krohne–Seward; both the continuous upper and lower bounds reduce to it.
  • domain assumption Borel toast theorem: existence of a Borel nested family of finite connected sets with large boundary separation (Theorem 6.2 / [4]).
    Used as the scaffolding for the Borel 5-coloring construction.
  • standard math There are exactly 35 non-isomorphic strong tournaments on 6 vertices, listed by score sequence and backward arcs.
    Classical enumeration (Moon, OEIS A051337); the lower-bound case analysis is exhaustive only relative to this list.
  • standard math Ergodicity of the free Z-action generated by e1-e2 on F(2^{Z^n}) for the measurable/Borel 4-color obstruction.
    Standard fact used in Proposition 6.1.
invented entities (2)
  • admissible energy function η (diamond-compatible + both energy subdigraphs coprime-free) no independent evidence
    purpose: Produces the long-tile obstruction that rules out 31 of the 35 strong 6-tournaments.
    Defined and verified inside §3; no external existence claim beyond the finite checks.
  • marker sets M_T ⊆ P2(T) for the four remaining tournaments TA,TB and reversals no independent evidence
    purpose: Produces the torus-tile obstruction via local transfer and row nontriviality.
    Explicit finite lists; transfer law checked case-by-case.

pith-pipeline@v1.1.0-grok45 · 24978 in / 3327 out tokens · 23024 ms · 2026-07-14T16:51:55.017009+00:00 · methodology

0 comments
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. And we prove that the Borel and measurable oriented chromatic number of directed Schreier graph $\vec F(2^{\mathbb Z^n})$, $n>1$ is 5.

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 ↗

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Reference graph

Works this paper leans on

11 extracted references · 1 linked inside Pith

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    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