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arxiv: 1907.11186 · v1 · pith:2SFCBST6new · submitted 2019-07-25 · 🧮 math.CO

Block-avoiding point sequencings of directed triple systems

Donald L. Kreher , Douglas R. Stinson , Shannon Veitch This is my paper

Pith reviewed 2026-05-24 16:01 UTC · model grok-4.3

classification 🧮 math.CO
keywords directed triple systemsv-good sequencingpoint sequencingsblock-avoiding permutationstransitive triplesdirected graph decomposition
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The pith

Every directed triple system order v congruent to 0 or 1 mod 3 admits a design with a v-good sequencing, but also admits designs without one when v is at least 7.

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

The paper proves that for every positive integer v congruent to 0 or 1 modulo 3 there exists a directed triple system DTS(v) equipped with a v-good sequencing, a permutation of the points in which no transitive triple appears with its elements in strictly increasing positions. It further proves that when v is at least 7 and satisfies the same congruence, there also exists a DTS(v) possessing no v-good sequencing at all. The work therefore establishes both the attainability of such order-avoiding permutations and their non-universality across all designs of a given order. Computational checks for all nonisomorphic DTS(v) with v at most 7 supply supporting data for the small cases.

Core claim

A v-good sequencing of a DTS(v) is a permutation of its v points such that for every transitive triple (x,y,z) in the design it is never the case that the indices satisfy i < j < k. The authors prove existence of a DTS(v) that admits such a sequencing for every v ≡ 0 or 1 mod 3, and existence of a DTS(v) that admits no such sequencing for every v ≡ 0 or 1 mod 3 with v ≥ 7.

What carries the argument

The v-good sequencing, defined as a permutation of the point set in which no block of the directed triple system occupies three positions in increasing order.

If this is right

  • For every admissible v there is at least one DTS(v) whose blocks can be ordered so that none appears in index-increasing order.
  • For every admissible v at least 7 there is at least one DTS(v) in which every possible permutation forces at least one block into increasing index order.
  • The computational enumeration for v ≤ 7 confirms that both kinds of designs occur among the nonisomorphic systems of those orders.

Where Pith is reading between the lines

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

  • The existence of both kinds of designs for each large admissible v suggests that the property of admitting a v-good sequencing is neither vacuous nor exhaustive.
  • One could ask whether the proportion of DTS(v) that admit v-good sequencings tends to a limit as v grows.
  • The constructions may extend to related decompositions such as directed quadruple systems or other oriented designs.

Load-bearing premise

Directed triple systems exist for every order v congruent to 0 or 1 modulo 3.

What would settle it

An order v ≡ 0 or 1 mod 3 for which every DTS(v) lacks a v-good sequencing, or for which every DTS(v) possesses one when v ≥ 7.

read the original abstract

A directed triple system of order $v$ (or, DTS$(v)$) is decomposition of the complete directed graph $\vec{K_v}$ into transitive triples. A $v$-good sequencing of a DTS$(v)$ is a permutation of the points of the design, say $[x_1 \; \cdots \; x_v]$, such that, for every triple $(x,y,z)$ in the design, it is not the case that $x = x_i$, $y = x_j$ and $z = x_k$ with $i < j < k$. We prove that there exists a DTS$(v)$ having a $v$-good sequencing for all positive integers $v \equiv 0,1 \bmod {3}$. Further, for all positive integers $v \equiv 0,1 \bmod {3}$, $v \geq 7$, we prove that there is a DTS$(v)$ that does not have a $v$-good sequencing. We also derive some computational results concerning $v$-good sequencings of all the nonisomorphic DTS$(v)$ for $v \leq 7$.

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 / 2 minor

Summary. The manuscript proves that for every positive integer v ≡ 0 or 1 (mod 3) there exists a directed triple system DTS(v) that admits a v-good sequencing (a permutation of the point set avoiding any forward-oriented transitive triple), and that for every such v ≥ 7 there also exists a DTS(v) with no v-good sequencing. Computational enumeration of all nonisomorphic DTS(v) for v ≤ 7 is reported, together with the status of their v-good sequencings.

Significance. The dual existence results give a sharp delineation of the ordering properties of directed triple systems, complementing the classical existence theorem for DTS(v) itself. The explicit constructions and non-existence arguments, together with the small-order exhaustive check, supply concrete, falsifiable information about block-avoiding permutations in decompositions of the complete digraph.

minor comments (2)
  1. The computational results for v ≤ 7 are mentioned in the abstract and presumably appear in a dedicated section or table; if the actual counts or isomorphism representatives are not displayed, a compact table listing the number of nonisomorphic DTS(v) and the fraction admitting a v-good sequencing would improve readability.
  2. Notation for the directed triple (x,y,z) and the sequencing [x1 … xv] is introduced clearly, but a short reminder of the transitive orientation convention at the beginning of the non-existence argument would help readers who consult only that section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were provided.

Circularity Check

0 steps flagged

No significant circularity; direct existence/non-existence proofs

full rationale

The paper establishes existence of DTS(v) with v-good sequencing via explicit constructions for v ≡ 0,1 mod 3, and non-existence via case arguments for v ≥ 7. No equations, fitted parameters, or predictions appear. No self-citations function as load-bearing uniqueness theorems; background existence of DTS(v) is standard external fact. Derivation chain is self-contained combinatorial argument without reduction to inputs by definition or citation loop.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard existence of DTS(v) for v ≡ 0,1 mod 3 and on the definition of transitive triples in the complete directed graph; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Directed triple systems DTS(v) exist for every v ≡ 0 or 1 mod 3.
    Invoked implicitly as the objects whose sequencing properties are studied.
  • standard math The complete directed graph can be decomposed into transitive triples.
    This is the definition of a DTS(v) and is a standard fact in design theory.

pith-pipeline@v0.9.0 · 5734 in / 1353 out tokens · 23902 ms · 2026-05-24T16:01:06.764203+00:00 · methodology

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Reference graph

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