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arxiv: 2605.20002 · v1 · pith:OVOX3B4Enew · submitted 2026-05-19 · 🧮 math.CO

Locally Semi-Equitable Colourings of BIBDs

Andrea C. Burgess , William Kellough , David A. Pike This is my paper

Pith reviewed 2026-05-20 03:29 UTC · model grok-4.3

classification 🧮 math.CO
keywords balanced incomplete block designcolouringHadamard matrixaffine planetwin prime powernecessary conditionscombinatorial design
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The pith

Colourings of BIBDs with one missing colour per block yield new necessary conditions for Hadamard matrices, affine planes, and twin prime powers.

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

The paper examines ℓ-colourings of balanced incomplete block designs in which each block omits exactly one colour and spreads the remaining colours evenly across its points. From this local balance requirement the authors derive several necessary arithmetic conditions on the design parameters v, k and λ. These conditions are then applied to produce fresh obstructions to the existence of Hadamard matrices, affine planes and twin prime powers.

Core claim

We study ℓ-colourings of (v,k,λ)-BIBDs where, within each block, one colour is absent and each of the ℓ-1 other colours appears exactly k/(ℓ-1) times. Several necessary conditions for such colourings to exist are established. These coloured BIBDs are used to provide new necessary conditions for the existence of Hadamard matrices, affine planes, and twin prime powers.

What carries the argument

The locally semi-equitable ℓ-colouring of a BIBD, which forces each block to omit one colour while balancing the frequencies of the others, thereby converting local uniformity into global divisibility constraints on the design parameters.

If this is right

  • Any BIBD admitting such a colouring must satisfy explicit divisibility conditions on its parameters.
  • Hadamard matrices of certain orders are ruled out by the new constraints obtained from coloured BIBDs.
  • Affine planes of particular orders become impossible once the colouring conditions are imposed.
  • Twin prime power constructions must obey additional arithmetic restrictions derived from the same colourings.

Where Pith is reading between the lines

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

  • The same local-balance technique might be applied to other families of designs to generate further existence obstructions.
  • The conditions can be checked by computer for small parameter sets, pruning impossible candidates before construction attempts begin.
  • If the necessary conditions turn out to be sufficient for some parameter ranges, they could support constructive existence proofs as well.

Load-bearing premise

The local per-block balance condition can be converted into global divisibility or integrality constraints on v, k and λ by ordinary double-counting or incidence-matrix arguments.

What would settle it

A specific (v,k,λ)-BIBD whose parameters violate one of the derived conditions yet still admits a locally semi-equitable colouring.

Figures

Figures reproduced from arXiv: 2605.20002 by Andrea C. Burgess, David A. Pike, William Kellough.

Figure 1
Figure 1. Figure 1: Π1 Π2 Π3 A1 = {0, 1} A3 = {0, 2} A5 = {0, 3} A2 = {2, 3} A4 = {1, 3} A6 = {1, 2} [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: A (4, 3, 2)-BIBD. A1 A2 A3 A4 A5 A6 A1 A2 A3 A4 A5 A6 A1 A2 A3 A4 A5 A6 A1 A2 A3 A4 A5 A6 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Combining points from the (4, 3, 2)-BIBD and the (4, 2, 1)-RBIBD. the square is paired with the resolution class Π1 = {A1, A2}, the pentagon is paired with the resolution class Π2 = {A3, A4}, and the hexagon is paired with the resolution class Π3 = {A5, A6}. We repeat this similarly for the other Di . For every {shape, resolution class} pair, we create two new points which are themselves pairs: {shape, Ai}… view at source ↗
Figure 3
Figure 3. Figure 3: The construction ends by taking each block in [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Forming transversal designs from the points constructed in Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The 16 blocks of a 0-ULSE 4-coloured (16 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The set of blocks Bm for the TD1(4, 3), Tm. where addition in the subscripts is done modulo 5. The blocks Bm of Tm are illustrated in [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
read the original abstract

We study $\ell$-colourings of $(v,k,\lambda)$-BIBDs (balanced incomplete block designs) where, within each block, one colour is absent and each of the $\ell-1$ other colours appears exactly $\frac{k}{\ell-1}$ times. We establish several necessary conditions for such colourings to exist. We also use these coloured BIBDs to provide new necessary conditions for the existence of Hadamard matrices, affine planes, and twin prime powers.

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 / 3 minor

Summary. The manuscript introduces locally semi-equitable ℓ-colourings of (v,k,λ)-BIBDs in which each block omits exactly one colour and distributes the remaining ℓ−1 colours equally (each appearing k/(ℓ−1) times). It derives several necessary conditions for the existence of such colourings via double-counting on the incidence structure and applies the resulting integrality constraints to obtain new necessary conditions on the parameters of Hadamard matrices, affine planes, and twin-prime-power constructions.

Significance. If the derivations hold, the work supplies concrete divisibility and integrality obstructions that can be used to rule out parameter sets for the three families of objects mentioned. The approach relies on standard BIBD relations (bk=vr, λ(v−1)=r(k−1)) and projection onto the known eigenspaces of the incidence matrix; these are appropriate tools for the field and the manuscript presents the counting arguments explicitly.

minor comments (3)
  1. [§3] The statement of the main necessary condition (presumably in §3 or §4) would be clearer if the precise double-counting identity or matrix equation used to obtain the divisibility constraint were displayed as a numbered equation.
  2. [Introduction] An explicit small example (e.g., the affine plane of order 3 or the unique (7,3,1)-BIBD) illustrating a locally semi-equitable colouring would help readers verify the definition before the general counting arguments.
  3. [§5] The application to twin prime powers in the final section would benefit from a short table comparing the new necessary condition with the classical Bruck–Ryser–Chowla condition for the same parameter sets.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The report recognizes the use of standard BIBD counting arguments and incidence-matrix eigenspace projections to obtain integrality obstructions, and notes the potential utility of these conditions for ruling out parameter sets in the three families of objects studied. No specific major comments or criticisms were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives necessary conditions for the existence of locally semi-equitable ℓ-colourings of (v,k,λ)-BIBDs by applying the local per-block coloring balance (one colour absent, others equally distributed) to the standard BIBD axioms via double-counting on the incidence structure or eigenvalue projections. These yield global divisibility/integrality constraints on v, k, λ directly from the definitions and the relations bk=vr, λ(v-1)=r(k-1). The subsequent restrictions on Hadamard matrices, affine planes, and twin prime powers are obtained by substituting the known parameter relations for those objects into the same constraints. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the argument is independent of any prior result by the same authors and relies only on classical design theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.0 · 5601 in / 1038 out tokens · 53513 ms · 2026-05-20T03:29:33.135249+00:00 · methodology

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