The 3-way flower intersection problem for Steiner triple systems

H. Amjadi; N. Soltankhah

REVIEW 2 major objections 4 minor 23 references

The 3-way flower intersection problem for Steiner triple systems

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Flower-sharing Steiner triple systems attain every allowed overlap

desk verdict New and mostly true result in a niche area, but the written proof leaves two load-bearing steps as assertions. read the letter →

arxiv 1908.06679 v2 pith:4252C7DU submitted 2019-08-19 math.CO

H. Amjadi , N. Soltankhah This is my paper · ORCID

classification math.CO MSC 05B0705B0505B15

keywords Steinertriplesystem3-wayintersectionflowerPairwisebalanceddesignGroupdivisibleLatinsquareproblemrecursiveconstruction

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

In a Steiner triple system on $2r+1$ points, the flower at a point is the set of $r$ triples through that point. This paper determines which numbers $k$ can occur as the number of additional common triples of three Steiner triple systems that already share one whole flower, for almost every admissible order $r$ (those with $r\equiv 0,1 \pmod 3$). For every $r\ge 10$ with $r\ne 24$, the achievable set $J^3_F(r)$ equals the counting bound $I^3_F(r)=S_3[2r(r-1)/3]$, meaning all values from $0$ up to the maximum except the six values immediately below it. The orders $r=3$ and $r=4$ are settled exactly, while $r=6,7,9,24$ are left with explicit gaps between lower and upper bounds. The result matters because it shows that no subtle obstruction beyond counting prevents three systems from sharing a prescribed number of blocks together with a full flower.

What carries the argument

The central mechanism is a family of recursive constructions that build $STS(2n+1)$ out of smaller Steiner triple systems plus Latin squares. Three systems are stacked on $V\times\{1,2,3\}$ and joined by Latin-square blocks (Constructions 1–3), or a system is doubled through a complete-graph $1$-factorization (Construction 4); independent control of the three inputs comes from the known 3-way intersection sets for Latin squares and for Steiner triple systems. A summation theorem (Theorem 8) then glues blocks of a pairwise balanced or group divisible design: if each block size $|B|$ admits a flower-intersection value $k_B$, the whole order admits $\sum_B k_B$. A transfer lemma (Lemma 4, with Corollary 1) turns large ordinary 3-way intersection numbers into flower intersection numbers by asserting that sufficiently large common intersections force a point whose flower is identical in all three systems. The named object $S_3[m]$ is the target set whose six top-end gaps recur throughout the recursion.

What would settle it

Test Lemma 4 by computer search for small admissible orders: for $r=6$ or $r=7$, look for three STS(13) or STS(15) whose common intersection has size at least $r+2r(r-3)/3$ but in which no point has an identical flower in all three systems. One such triple of systems would break the transfer step that Corollary 1 uses for every $r\ge 9$; alternatively, proving that some $k\in I^3_F(n)$ with admissible $n\ge 10$, $n\ne 24$ cannot be realized would directly contradict Theorem 20.

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Extended reading notes

Core claim

The main theorem states that for $n\equiv 0,1 \pmod 3$, $n\ge 10$, $n\ne 24$, the set $J^3_F(n)$ of 3-way flower intersection numbers coincides with $I^3_F(n)=S_3[2n(n-1)/3]$, where $S_3[m]$ is all nonnegative integers $\le m$ except $m-1,m-2,m-3,m-4,m-5,m-7$. It also gives the complete small cases $J^3_F(3)=\{4\}$ and $J^3_F(4)=\{0,8\}$, and for $r=6,7,9,24$ it proves containment bounds that leave only short lists of values undecided. The proof constructs, for every claimed $k$, three explicit Steiner triple systems with exactly one common flower and exactly $k$ further common triples. The equality says that the full three-way analogue of the classical flower intersection problem has the same shape as the ordinary three-way intersection problem: a full interval with six top-end exceptions.

Load-bearing premise

The load-bearing premise is Lemma 4: whenever three Steiner triple systems of order $2r+1$ share at least $r+2r(r-3)/3$ triples, some point has the same flower in all three systems. The paper supports this with a one-sentence 'simple calculation' sketch, and Corollary 1 plus every large-order case built from it depends on that calculation being correct.

Editorial extensions

If this is right

All admissible orders $n\ge 10$ except $n=24$ have $J^3_F(n)=I^3_F(n)$, so every allowed value from $0$ to $2n(n-1)/3$, apart from the six top-end exceptions, is realized by three flower-sharing systems.
The undecided orders are exactly $n=6,7,9,24$, with only short explicit value lists left open: for instance $k=6$ for $n=6$ and $k\in\{9,14,15\}$ for $n=7$.
The induction rests on small bases $n=3,4,10,15,60,132$; once these are available in the right residue classes, the PBD/GDD gluing produces every larger case.
The maximum value $2n(n-1)/3$ is always realized by taking three identical systems, and $0$ is realized for all $n\ge 4$, so the extreme ends of the spectrum are never exceptional.

Reading between the lines

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

The recurring six-gap shape $S_3[m]$ across solved orders suggests that for a $\mu$-way flower intersection problem the target set may be $S_\mu[m]$, with the same style of top-end exceptions; the recursion here gives a template for proving such a general statement.
The unresolved cases are narrow enough that a finite exhaustive search could close them: only one value ($k=6$) is missing for $r=6$, and only $\{9,14,15\}$ for $r=7$, given the paper's bounds.
The same Latin-square stacking and PBD summation machinery should transfer to flower intersection problems for other designs, such as $S(2,4,v)$ or Kirkman triple systems with a prescribed parallel class, where analogous ingredients are known.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper mostly settles the 3-way flower intersection problem for Steiner triple systems: for almost all admissible orders it pins down J3F(r), and that set was previously unknown. The result is new and, as far as I can tell, true. The machinery is standard—Latin square constructions, PBD/GDD blow-ups, small-case checking—but the assembly is genuine work, and the small cases are listed in enough detail to be checked. The citations to Milici-Quattrocchi, Hoffman-Lindner, Adams et al., and the authors' earlier KTS paper all look appropriate; I see no circularity.

Two soft spots keep me from calling the proof complete. The first is Lemma 4, where the averaging argument is compressed into one sentence. The stress-test note's sketch—if no point had a common flower, each point would lie in at least two noncommon triples in every system, forcing too many noncommon triples—is credible, so I doubt the lemma is false. But the proof as printed is not a proof. The second, more serious gap is the Main Theorem's recursive coverage step. The five cases say that the required objects are guaranteed and then stop; they never show that the sumsets produced by Theorems 2-8 actually cover every integer in I3F(n). This matters because the input sets for r=6,7,9 are only partially known. For n=42, for example, case 5 uses a B({6,7},1,42) and needs 49·J3F(6)+6·J3F(7) to fill all values below the Corollary 1 threshold. Since J3F(6) is not fully determined, the interval coverage is not immediate. It is probably still true, but the burden is on the authors to show the arithmetic or provide code. As written, the central equality is conditional on an unstated finite check.

A minor note: the small-case computer checks are attributed to helpers but no code or output is included. That is common in this area and not disqualifying, but it does mean those values are also taken on faith.

Who gets value from this paper: anyone working on STS intersection problems or flower intersections generally. It will be the reference for J3F(r), once the gaps are patched. It deserves a serious referee, not a desk rejection. I would send it out with a request for two things: a real proof of Lemma 4 (even a paragraph), and an explicit verification—table or code—that the recursive constructions cover the required intervals. If those land, the paper is solid.

Referee Report

2 major / 4 minor

Summary. The paper studies the 3-way flower intersection problem for Steiner triple systems. For admissible r (r ≡ 0, 1 mod 3), J3F(r) is the set of integers k for which there exist three STS(2r+1) whose total common triples consist of k+r triples, r of them forming a common flower. The main theorem (Theorem 20) claims that J3F(n) = I3F(n) for every admissible n ≥ 10 with n ≠ 24, with partial information for n = 6, 7, 9, 24. The proof combines four recursive constructions (Latin-square constructions, a doubling construction, and PBD/GDD-based constructions), a lemma asserting that sufficiently large 3-way intersections force a common flower, and a collection of explicitly listed small base cases, some of which are checked by computer.

Significance. If the result is correct, it gives a complete solution to a natural 3-way analogue of the Hoffman–Lindner flower-intersection problem and extends the Milici–Quattrocchi 3-way intersection theorem to flowers. The paper has a clear recursive framework and provides a large amount of explicit small-case data, which is a genuine strength. However, two load-bearing parts of the proof are not actually demonstrated in the manuscript: Lemma 4 is justified by a one-sentence handwave, and the main theorem's recursive cases assert rather than verify the necessary set-arithmetic coverage. These gaps make the central equality J3F(n) = I3F(n) unverifiable from the proof as written, so the manuscript needs a substantial revision before it can be accepted.

major comments (2)

[§2, Lemma 4] Lemma 4 is load-bearing: Corollary 1, and through it nearly all high-end values in the small cases and in the main theorem, are derived from this lemma. The proof, however, consists only of the sentence 'A simple calculation now shows that there must be at least one point x for which the triples through x are the same in all systems.' No calculation is given, and the argument does not explicitly account for triples common to two of the three systems but not to the third. The lemma is very likely correct, and a counting proof is available: the condition k ≥ 2r(r−3)/3 makes the total number of non-common triple occurrences across the three systems at most 3·(4r/3) = 4r, whereas if no point has a common flower, each of the 2r+1 points is contained in at least one non-common triple in each of the three systems, giving at least 3(2r+1) = 6r+3 occurrences. This argument should be written out in full, or the lemma should be replaced by a reference to a complete proof.
[§4, proof of Theorem 20] The proof of the main theorem asserts, in each of the five recursive cases, that 'all required objects ... is guaranteed', but it never demonstrates that the sumsets produced by Theorems 2–8 actually cover every element of I3F(n) below the threshold of Corollary 1. This is a nontrivial arithmetic claim, especially because the input sets for r = 6, 7, 9 are only partially known. For example, in Case 5 with n = 42 (t = 7), Theorem 8 must combine 49 copies of the partial set J3F(6) ⊇ [0,5] ∪ {7,20} and 6 copies of the partial set J3F(7) ⊇ [0,8] ∪ [10,13] ∪ {16,22,28} to cover all values below 1092; the paper gives no interval-covering lemma, no explicit sumset computation, and no code for this check. Similarly, Case 1 with n = 19 and Case 2 with n = 18, 21, 27 rely on partial input sets. These checks are load-bearing: if some integer in the required interval is not covered, the equality J3F(n) = I3F(n) would fail for that n. I am not claiming the theorem is false; spot checks suggest the coverage is likely verifiable, but the proof as written does not contain the required verification.

minor comments (4)

[§3, Theorems 12–14] The phrase 'It is checked by computer programming' should be supplemented with code or a machine-readable data file, or at least with a precise description of the verification procedure. The explicit systems and permutations are useful, but they are not sufficient for independent reproducibility without knowing the script used.
[§2, Theorem 1] The proof of Theorem 1 says the construction is 'exactly the same' as Lemma 2.2 of Adams et al. Please provide enough detail to show how the secondary diagonal is preserved and how the parameter b is counted, since Theorem 2 depends on the exact value of b.
[§2, Lemmas 5–7] The definition of 'special' Latin square and the description of the cells 'above the 2×2 diagonal blocks' is informal; precise row and column indexing would make the counts in K and M easier to check.
[§4, proof of Theorem 20] There are several small presentation issues: 'All required objects ... is guaranteed' should be 'are guaranteed', and 'F orr' at the start of Theorem 4 is a typo. More importantly, the proof should state explicitly how Theorem 1 supplies the Latin-square parameter b in Cases 1 and 3.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the claimed equality is obtained from external intersection theorems and constructive recurrences, not from a definitional identification of J3F with I3F.

full rationale

The paper does not fit any parameter or define I3F in terms of J3F. Lemma 1 gives only necessary inclusions; the equality direction is built by explicit constructions. Theorems 2-5 and 8 map known J3F values of smaller admissible orders into larger ones, and Corollary 1 supplies the upper range via Milici-Quattrocchi's independent J3(v)=I3(v) theorem and Lemma 4; no step equates an output to its input by construction. The use of Amjadi-Soltankhah (2017) in Theorem 13 is a self-citation, but it supplies concrete KTS(15) systems whose flower intersections are externally checkable, and the cited KTS result is not an input to the present flower-intersection equality; hence it is not circular. The main proof does contain terse assertions of coverage ('All required objects ... is guaranteed') and a one-sentence sketch in Lemma 4; these are rigor and completeness concerns, not circularity, because the missing arithmetic is a verification task rather than a definitional equivalence.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claim rests on standard existence theorems for STS, KTS, PBDs, GDDs, MOLS, and Latin square intersections, all taken from the cited literature. It also rests on two paper-specific assertions: the correct execution of computer checks for small r, and the unstated set-covering step in the main recursive proof. No free parameters are fitted; no new entities are introduced.

assumptions (9)

domain assumption There exists an STS(v) iff v ≡ 1,3 (mod 6).
Used throughout; cited to Hanani (1975).
domain assumption J3(v)=I3(v) for v≥19, with the listed small values (Theorem A of Milici-Quattrocchi).
Used in Lemma 4 application, Corollary 1, and base cases.
domain assumption J'3(n) is known for Latin squares of every order (Theorem C of Adams et al.).
Used in Theorems 2-4 and small case constructions.
domain assumption JF(r)=IF(r) for admissible r (Theorem B of Hoffman-Lindner).
The 2-way flower intersection result that motivates this paper and provides boundary values.
domain assumption Four mutually orthogonal Latin squares exist for n≥5, n≠6, except possibly n=10,18,22 (Colbourn-Dinitz, Todorov).
Used in Theorem 6 to build the PBD of order 6t.
domain assumption Specific GDDs exist: {4}-GDD of type 3^4 6^2, 6^10, and 9^12 24^1 (Kreher-Stinson, Wei-Ge).
Used in Theorems 17-19 for r=24,60,132.
domain assumption Large sets of disjoint 3-GDDs of type 2^r exist for admissible r (Cao, Lei, Zhu).
Used in Lemma 2 to show 0 is in J3F(r) for r≥4.
ad hoc to paper The computer-verified intersection numbers in Theorems 12-14 are correct.
No code supplied; the paper says 'It is checked by computer programming'.
ad hoc to paper The recursive constructions in the Main Theorem cover every integer in I3F(n) for the five cases.
The proof does not show the interval-covering arithmetic.

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Cite this review

Pith. "Pith review of The 3-way flower intersection problem for Steiner triple systems." pith.science (2026). https://pith.science/paper/4252C7DU

@misc{pith2026190806679,
  author       = {Pith},
  title        = {Pith review of: The 3-way flower intersection problem for Steiner triple systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4252C7DU}},
  note         = {Machine review of arXiv:1908.06679}
}

read the original abstract

The flower at a point x in a Steiner triple system (X; B) is the set of all triples containing x. Denote by J3F(r) the set of all integers k such that there exists a collection of three STS(2r+1) mutually intersecting in the same set of k + r triples, r of them being the triples of a common flower. In this article we determine the set J3F(r) for any positive integer r = 0, 1 (mod 3) (only some cases are left undecided for r = 6, 7, 9, 24), and establish that J3F(r) = I3F(r) for r = 0, 1 (mod 3) where I3F(r) = {0, 1,..., 2r(r-1)/3-8, 2r(r-1)/3-6, 2r(r-1)/3}.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Adams, E

P. Adams, E. J. Billington, D. E. Bryant, and E. S. Mahmoodian. The 3 -way intersection problem for L atin squares. Discrete Mathematics, 243 0 (1–3): 0 1--19, 2002

work page 2002

[2] [2]

Amjadi and N

H. Amjadi and N. Soltankhah. The 3 -way intersection problem for K irkman triple systems. Graphs and Combinatorics, 33: 0 673--687, 2017

work page 2017

[3] [3]

E. J. Billington, D. M. Donovan, and N. Smith. Packing a complete graph of order 5 (mod 6 ) with triangles: a new method. Bulletin of the Institute of Combinatorics and its Applications, 53: 0 77--81, 2008

work page 2008

[4] [4]

H. Cao, J. Lei, and L. Zhu. Large sets of disjoint group divisible designs with block size three and type 2^n4^1 . Journal of Combinatorial Designs, 9 0 (4): 0 285--296, 2001

work page 2001

[5] [5]

Chang and G

Y. Chang and G. Lo Faro. Intersection numbers of K irkman triple systems. Journal of Combinatorial Theory, Series A, 86 0 (2), 1999

work page 1999

[6] [6]

Chang and G

Y. Chang and G. Lo Faro. The flower intersection problem for K irkman triple systems. Journal of Statistical Planning and Inference, 110 0 (1–2): 0 159--177, 2003

work page 2003

[7] [7]

C. J. Colbourn and J. H. Dinitz. Handbook of combinatorial designs. CRC press, 2006. ISBN 1420010549

work page 2006

[8] [8]

Golalizadeh and N

S. Golalizadeh and N. Soltankhah. On the existence of d -homogeneous -way (v,3,2) trades. Graphs and Combinatorics, 35: 0 471--478, 2019

work page 2019