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On the number of small Steiner triple systems with Veblen points

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Pith's one-line read The paper establishes exact counts of Steiner triple systems with Veblen points at orders 19, 27 and 31.

desk verdict First exact counts for STS(19), (27), and partial (31) with Veblen points; credible for small orders, but the order-31 table rests on an unverified 2^35 enumeration. read the letter →

arxiv 2411.16307 v2 pith:YPT5M7Y3 submitted 2024-11-25 math.CO math.GR

Mario Galici , Giuseppe Filippone This is my paper · ORCID
classification math.COmath.GR MSC 05B0720N0551E10
keywords SteinertriplesystemsVeblenpointsSchreierextensionsloopscentralSTS(19)STS(27)STS(31)
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes exact isomorphism-class counts for Steiner triple systems of small orders that contain Veblen points — points through which any two triples generate a Pasch configuration. It proves that exactly three of the 11,084,874,829 non-isomorphic STS(19)s have one Veblen point, that exactly 1,736 non-isomorphic STS(27)s have one Veblen point, and that exactly two non-isomorphic non-projective STS(31)s have three Veblen points. It also gives exact counts, for six selected quotient systems, of STS(31)s with exactly one Veblen point. These are complete counts within the corresponding subclasses, not asymptotic estimates. The interest is that counting all Steiner triple systems beyond order 21 is open, while Veblen-point-bearing systems are regular enough that a loop-theoretic construction can enumerate them exactly.

What carries the argument

The machinery is the Schreier extension of Steiner loops. Every Steiner triple system gives a Steiner loop on S ∪ {Ω}, and Veblen points are exactly the nontrivial central elements of that loop. Starting from a central elementary abelian 2-group L_N and a quotient Steiner loop L_Q, every extension is encoded by a symmetric factor system f : L_Q × L_Q → L_N that satisfies a prescribed condition on the triples of Q; the total number of factor systems is |L_N|^b where b is the number of triples of Q. Two extensions are equivalent when they differ by a co-boundary, and when L_N is the full center, isomorphism classes are the orbits of Aut(L_N) × Aut(L_Q) on non-equivalent factor systems. The extra-Veblen criterion of Remark 1.1 decides, via a linear condition on f, whether an element outside N becomes central, and this filter is what separates the STS(31) cases where the quotient itself has Veblen points.

What would settle it

Search the published complete classification of the 11,084,874,829 STS(19)s for Veblen points: if the number of isomorphism classes with exactly one Veblen point is not three, Theorem 2.1 fails. For the STS(31) counts, re-run the factor-system enumeration with an independently written program; a single discrepancy in Table 4 or in the two three-Veblen-point systems would falsify the corresponding theorem.

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Extended reading notes

Core claim

The central discovery is that the Schreier-extension machinery for Steiner loops yields exact global counts in small orders where a full classification is infeasible. Concretely, Theorem 2.1 identifies the unique three STS(19)s with exactly one Veblen point; Theorem 3.1 gives 1,504 plus 232, hence 1,736, STS(27)s with exactly one Veblen point, split by whether the quotient STS(13) is non-cyclic or cyclic; Theorem 4.2 gives exactly two non-projective STS(31)s with exactly three Veblen points, described by explicit factor systems f1 and f2; and Theorem 4.1 with Table 4 gives exact STS(31) one-Veblen-point counts for six chosen quotient STS(15)s, from 278 for PG(3,2) to 99,952 for STS(15)#61.

Load-bearing premise

The counts rest on the assumption that every Steiner triple system with the prescribed number of Veblen points arises as a Schreier extension of its full center by its quotient, and that the isomorphism classes of these systems are exactly the automorphism orbits of factor systems described in the paper, with the paper's criterion correctly identifying all additional Veblen points.

Editorial extensions

If this is right

  • Any complete classification of STS(19) must contain exactly the three systems described in Theorem 2.1 as its Veblen-point-bearing members.
  • Any full enumeration of STS(27) must include exactly 1,736 systems with a single Veblen point: 1,504 with the non-cyclic STS(13) quotient and 232 with the cyclic one.
  • For order 31, apart from PG(4,2), exactly two non-isomorphic systems have three Veblen points, so the three-Veblen-point family is completely known.
  • The six counts in Table 4 pin down, exactly, how many STS(31)s over specific quotients have one Veblen point; for example, only 278 arise over PG(3,2) despite 2^24 non-equivalent extension classes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same orbit-counting recipe could be applied to order 21, where no complete classification exists, to obtain exact counts for the one-Veblen-point subfamily and thereby a lower bound on the total number of STS(21)s.
  • The three-Veblen-point count being only two suggests that non-projective systems with many Veblen points may be rare; a testable conjecture is that among non-projective STS(v) the number with at least c Veblen points grows far slower than the total number of systems.
  • The Appendix describes a batch computation over 2^35 factor systems, so an independent reimplementation on different hardware would directly confirm the six values in Table 4.
  • Because the two STS(31)s with three Veblen points are presented only through factor systems, an explicit 155-triple listing could be extracted and checked for additional properties, such as which subsystems of order 7 or 9 they contain.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper uses the Schreier-extension theory of Steiner loops, developed by the authors in a previous paper [6], to enumerate Steiner triple systems of orders 19, 27, and 31 that contain Veblen points. The main results are: (i) exactly 3 non-isomorphic STS(19)s have a unique Veblen point, and explicit factor systems are given; (ii) exactly 1736 non-isomorphic STS(27)s have a unique Veblen point, split as 1504 with the non-cyclic STS(13) as quotient and 232 with the cyclic STS(13) as quotient; (iii) exactly 2 non-isomorphic non-projective STS(31)s have exactly three Veblen points; (iv) for six selected quotients Q of order 15, Table 4 lists the numbers of STS(31)s with a unique Veblen point and quotient Q, obtained by enumerating 2^35 factor systems per quotient and reducing by coboundaries and automorphism orbits.

Significance. If correct, these are exact isomorphism-class counts for subclasses that have not been enumerated before, and they demonstrate the reach of the loop-extension method for STS. The paper provides explicit coordinate representations for the three STS(19)s and the two STS(31)s with three Veblen points, and it makes its source code publicly available, which are concrete strengths. The counts are, however, contingent on two external supports: the classification theorem from [6] that identifies isomorphism classes with automorphism orbits, and a large batch computation (2^35 factor systems per quotient) for the order-31 case that is not independently certified in the manuscript.

major comments (3)
  1. [Section 4, Theorem 4.1 and Table 4] The theorem statement is broader than the result actually proved: it claims that the number of non-isomorphic STS(31)s with exactly one Veblen point and quotient system Q of order 15 is given by Table 4, but Table 4 lists only six of the 80 non-isomorphic STS(15)s, and the proof explicitly restricts to those six. Please restate the theorem as 'for the quotient systems listed in Table 4' or otherwise qualify it, otherwise the reader may conclude that a complete classification for order 31 has been obtained.
  2. [Section 1.1, Eq. (2) and Remark 1.1] The central reduction that isomorphism classes of extensions are exactly the orbits of Aut(LN) × Aut(LQ) on non-equivalent factor systems when LN is the whole center, as well as the extra-Veblen criterion of Remark 1.1, are imported from [6] without proof or a precise statement of the hypotheses. These are load-bearing in Theorems 2.1, 3.1, 4.1, and 4.2. If the theorem in [6] requires conditions (e.g., that the action is well-defined on cohomology classes, or that every STS with the given center arises from such a factor system), the applications here need to be checked against those conditions. Please include a formal statement of the imported results (or a proof sketch) and verify the hypotheses explicitly for each application.
  3. [Appendix A.1 and Theorem 4.1] The order-31 counts in Table 4 rest on an enumeration of 2^35 factor systems for each of the six quotients, and the paper provides no independent certificate or fully reproducible algorithmic specification. The appendix describes the encoding of factor systems as integers, the computation of coboundaries, and a batch approach, but it does not give pseudo-code for the orbit computation or the extra-Veblen filter, nor does it report checks against known counts on smaller orders. Since a bug in the encoding of fundamental pairs, the induced permutation on the 35-bit representation, the orbit computation, or the filter would change every entry of Table 4, the authors should supply a stronger correctness argument: for example, reproductions of the orbit counts for the order-19 and order-27 cases using the same code, or independent verification of Table 4 by a second implementation.
minor comments (5)
  1. [p. 4, first paragraph of Section 1] The word 'Setiner' is a typo and should be 'Steiner'.
  2. [Section 4, proof of Theorem 4.2] The statement '4^7 = 16383' is arithmetically incorrect; 4^7 = 16384. The subsequent division by |B^2| = 256 is consistent with the correct value, so this is a typo, but it should be fixed.
  3. [Section 4, proof of Theorem 4.2] The sentence 'these orbits do not necessarily coincide with the isomorphism classes of STS(19)s with three Veblen points' should refer to STS(31)s, not STS(19)s.
  4. [Table 3 caption] The caption reads 'The STS(19) S1 with unique Veblen point 0', but the table is for S2, as the text immediately before it explains.
  5. [Appendix A.1, paragraph on automorphisms] In the sentence 'Let us now consider a factor systems f represented as a vector ... and two automorphisms α ∈ Aut(LN), β ∈ Aut(LN)', the second automorphism should be β ∈ Aut(LQ), since it acts on the quotient coordinates.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction; the counts are genuine orbit enumerations, though the method leans on the authors' prior algebraic framework from [6].

full rationale

The paper's numerical assertions (3 STS(19)s, 1736 STS(27)s, Table 4 for quotient systems of order 15, and 2 STS(31)s with three Veblen points) are not obtained by fitting a parameter to a target value or by defining a quantity in terms of itself. Each count is produced by enumerating Schreier extensions: factor systems are reduced modulo coboundaries, then the action of Aut(LN) x Aut(LQ) is applied, and the resulting orbit counts are interpreted as isomorphism-class counts via the theorem quoted from [6]. No target count appears among the inputs of the computation, and the paper provides explicit factor systems and, for the small cases, explicit reconstructions of the Steiner triple systems. The load-bearing identifications (isomorphism classes of factor systems correspond to orbits when LN is the whole center; the extra-Veblen criterion in Remark 1.1; the index-at-most-4 projectivity theorem, Theorem 1.1) are quoted from [6], which shares an author with the present paper. These are genuine self-citations and are central to the method, but they are general algebraic theorems whose assumptions do not include the particular orders 19, 27, 31 or the enumerated totals; they are therefore independent support rather than circular premises. The 2^35-factor-system computation for STS(31)s is a substantial computational claim, and the appendix's note that a naive store would require about 64 PB, together with the absence of an independent second implementation, is a correctness risk but not a circularity risk. No equation in the paper exhibits a derivation that is equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the extension-theoretic framework from [6] and on the isomorphism classification of STS(9), STS(13), and STS(15). There are no fitted parameters, and no new entities are introduced. The main burden is that several load-bearing structural theorems are cited rather than proved in this paper.

assumptions (5)
  • standard math Known isomorphism classes of the STS(9), STS(13), and STS(15) quotients: unique STS(9), two STS(13)s, 80 STS(15)s.
    Taken from the CRC Handbook [3] and from [12]; used to fix the quotient systems in Sections 2-4.
  • domain assumption Veblen points are exactly the non-trivial central elements of the associated Steiner loop.
    Used throughout Section 1.1 and in the proofs; cited to [6].
  • domain assumption Central extensions of Steiner loops are exactly Schreier extensions, and isomorphism classes of extensions with full center correspond to orbits of Aut(LN) x Aut(LQ) on non-equivalent factor systems.
    Core counting principle in Theorems 2.1, 3.1, 4.1, and 4.2; cited to [6] and [2].
  • domain assumption Condition (3) in Remark 1.1 detects Veblen points outside N.
    Used to filter out factor systems for order-31 systems in Theorems 4.1 and 4.2; cited to [6].
  • domain assumption Theorem 1.1: an extension of index at most 4 yields a projective STS; non-projective STS(v) has at most (v-7)/8 Veblen points.
    Used in Section 4 to restrict the possible number of Veblen points in STS(31).

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Pith. "Pith review of On the number of small Steiner triple systems with Veblen points." pith.science (2026). https://pith.science/paper/YPT5M7Y3

@misc{pith2026241116307,
  author       = {Pith},
  title        = {Pith review of: On the number of small Steiner triple systems with Veblen points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YPT5M7Y3}},
  note         = {Machine review of arXiv:2411.16307}
}
read the original abstract

The concept of Schreier extensions of loops was introduced in the general case in [11] and, more recently, it has been explored in the context of Steiner loops in [6]. In the latter case, it gives a powerful method for constructing Steiner triple systems containing Veblen points. Counting all Steiner triple systems of order v is an open problem for v>21. In this paper, we investigate the number of Steiner triple systems of order 19, 27 and 31 containing Veblen points and we present some examples.

Figures

Figures reproduced from arXiv: 2411.16307 by the authors.

Figure 1
Figure 1. A Pasch configuration through a Veblen point [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The STS(9) as the affine plane AG(2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. provides a visual representation of the non-trivial factor systems f1 and f2, where the triples of the STS(9) Q in which fi is non-zero, i = 1, 2, are drawn in red. P1 P4 P7 P2 P8 P3 P6 P9 P5 P1 P4 P7 P2 P8 P3 P6 P9 P5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The Pasch switch transforming the triples of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: The Pasch switch transforming the triples of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: The Fano plane. The three orbits that we obtained are represented, respectively, by the following three factor systems: f0, the null function, i.e., the function mapping every pair into Ω; f1, defined by f1(P3, P5) = f1(P3, P6) = f1(P5, P6) = (0, 1), and Ω elsewhere; f…
Figure 7
Figure 7. Figure 7: provides a visual representation of the non-trivial factor systems f1 and f2, where the triples of Q in which fi is equal to (0, 1) are drawn in red, and the ones in which fi is equal to (1, 0) are drawn in blue, i = 1, 2. P4 P3 P7 P1 P2 P5 P6 P4 P3 P7 P1 P2 P5 P6 [PI…

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Works this paper leans on

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