New constructions and bounds for nonabelian Sidon sets with applications to Tur\'an-type problems

John Byrne; Michael Tait

REVIEW 3 major objections 3 minor 42 references

New constructions and bounds for nonabelian Sidon sets with applications to Tur\'an-type problems

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that the largest $S_k$-set in $S_n$ attains the trivial upper bound $(n!)^{1/k}$ up to $O(1/\log n)$, and uses these constructions to settle directed Turán-type problems.

desk verdict Theorem 3's printed statement is a trap, but the exponent-level claim survives; the digraph work is the genuinely valuable part. read the letter →

arxiv 2509.07750 v1 pith:AHKETZXT submitted 2025-09-09 math.CO

John Byrne , Michael Tait This is my paper · ORCID

classification math.CO MSC 05D0505C3505C2020B3005D40

keywords nonabelianSidonsetsS_k-setssymmetricgrouppermanentsCayleygraphsextremaldigraphtheoryminimumsemidegreeHamiltonpaths

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

This paper is trying to establish that the naive counting upper bound for nonabelian Sidon sets is often tight, and that these sets are a usable tool for extremal graph theory. Its headline result is that for every fixed $k$, the largest $S_k$-set in the symmetric group $S_n$ has size $(n!)^{1/k} + O(1/\log n)$, matching the trivial bound obtained by counting $k$-fold products. It also constructs $S_2$-sets of size $(n-1)!$ in $S_n \times S_n$ and $(n-1)!/2$ in $A_n \times A_n$, gives probabilistic lower bounds for $S_2$-sets and $S_2'$-sets in 'nice' groups, and proves upper bounds that improve the trivial constant when the group has a normal abelian subgroup of bounded index. On the graph side, the paper determines up to a constant factor the minimum semidegree that forces two distinct directed walks of length $k$ with the same endpoints, shows that forbidding a single $C_{\ell,\ell}$ forces $\Theta(n^{1/2})$ in this minimum-degree sense, improves the upper bound on Hamilton paths that create two-part cycles, and disproves a directed version of the Erdős-Simonovits compactness conjecture.

What carries the argument

The machinery has three parts. First, the lifting construction: given an $S_k$-set $A$ in a finite group $\Gamma$, the permutations $\pi$ of $\Gamma$ with $\pi(x)\in xA$ for all $x$ are themselves an $S_k$-set in $S_\Gamma$, and their number is the permanent of the 0-1 matrix $M_{x,y}=1$ if and only if $x^{-1}y\in A$. The permanent lower bound for doubly stochastic matrices, applied to $M/a$, is the estimate that makes the count reach $(n!)^{1/k+o(1)}$ after choosing $|\Gamma|$ near $n$ using primes in short intervals. Second, the conjugacy recipe: for a fixed $\pi$, the set $\{(\alpha,\alpha\pi): \alpha\in A\}$ in $\Gamma\times\Gamma$ is an $S_2$-set with parameter $g$ whenever every element of $\Gamma$ has at most $g$ preimages under the conjugation map $\alpha\mapsto\alpha\pi\alpha^{-1}$; point stabilizers in $S_n$ make $g=1$ or small, yielding exact factorial-sized sets. Third, for the digraph results, a random partition of vertices into $2k$ parts keeps only edges from part $i$ to part $i+1$, producing a spanning subgraph with a constant fraction of the minimum degree in which every closed walk of length at most $2k-1$ is balanced; this removes short unbalanced cycles and makes the remaining graph $F_k$-free.

What would settle it

For $k=2$ and a concrete seed $S_2$-set of size $a$ in a group of order $n=(p^2-1)2$, compute the permanent of the incidence matrix $M$; since this permanent is exactly the number of permutations produced by the construction, an exponential gap between $\operatorname{per}(M)$ and $a^n n!/n^n$ as $n$ grows would make the constructed family too small to match $(n!)^{1/k}+O(1/\log n)$.

Watch

Extended reading notes

Core claim

Stated on the paper's own terms, the central discovery is that the trivial product-counting bound $M_k(\Gamma) \le |\Gamma|^{1/k}$ can be asymptotically attained in the symmetric group: $M_k(S_n) = (n!)^{1/k} + O(1/\log n)$. The proof lifts an $S_k$-set $A$ in a smaller group $\Gamma$ of order near $n$ to the set of all permutations of $\Gamma$ that send each element $x$ into $xA$; the number of such permutations is a permanent of the 0-1 incidence matrix of $A$, and the permanent lower bound for doubly stochastic matrices is used to estimate it. A second, exact construction produces $S_2$-sets in $\Gamma \times \Gamma$ from a conjugacy class: if at most $g$ conjugates of $\pi$ by elements of $A$ can coincide, then $\{(\alpha,\alpha\pi): \alpha\in A\}$ is an $S_2$-set allowing at most $g$ words per product, and taking $A$ to be a point stabilizer in $S_n$ yields sets of size $(n-1)!$ in $S_n \times S_n$ and $(n-1)!/2$ in $A_n \times A_n$. The same families are then fed into Cayley graphs to show that the minimum semidegree forcing the directed subgraph family $F_k$ is $\Theta(n^{1/k})$, that a single $C_{\ell,\ell}$ forces $\Theta(n^{1/2})$, and that the directed compactness conjecture fails.

Load-bearing premise

The proof of the headline lower bound counts the constructed permutations by the permanent of a 0-1 matrix and relies on that count being within a subexponential factor of $a^n$, where $a$ is the size of the seed $S_k$-set; if the true maximum permanent for such regular matrices is exponentially smaller, the counting step cannot deliver $(n!)^{1/k}$.

Editorial extensions

If this is right

For every fixed $k$, the largest $S_k$-set in $S_n$ has size $(n!)^{1/k}+O(1/\log n)$, so the trivial product-counting bound is asymptotically exact in the symmetric group.
$S_2$-sets of size $(n-1)!$ exist in $S_n\times S_n$, and of size $(n-1)!/2$ in $A_n\times A_n$; these are within a factor of $n$ of the trivial bound $n!$ for the product group.
The minimum semidegree that guarantees two distinct directed walks of length $k$ with the same endpoints lies between $(1/(k^{1+1/k})-o(1))n^{1/k}$ and $(2k+o(1))n^{1/k}$; for a single $C_{\ell,\ell}$, the range is between $(1/(2\ell-2)^{1/2}-o(1))n^{1/2}$ and $(2\ell+o(1))n^{1/2}$.
For even $\ell\ge 4$, the maximum number of Hamilton paths on $[n]$ no two of which create a two-part cycle of length $\ell$ is at most $(n!)^{1/2+O(1/\log n)}$.
The directed analogue of the Erdős-Simonovits compactness conjecture is false: for the family $C_{k,k}$ of two internally disjoint directed paths, the ratio $m_0(n,H)/m_0(n,C_{k,k})$ tends to infinity for every member $H$.

Reading between the lines

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

A testable extension the authors do not pursue is to use the conjugacy recipe in other permutation groups with a large conjugacy class, such as wreath products; Proposition 3 already shows the mechanism transfers whenever a large conjugacy class exists, so the factorial-size phenomenon may be more general than $S_n$.
The permanent-counting step is the sensitive spot of Theorem 3; sharper estimates for permanents of regular 0-1 matrices, or a different counting argument, would either certify the $O(1/\log n)$ error or force a smaller lower bound, and this is the first place to look if the theorem is pushed to other groups.
Comparing Theorems 7 and 8 suggests an interpolation question: what is the smallest family of orientations whose minimum-semidegree Turán function passes from $\Theta(n^{1/k})$ to $\Theta(n^{1/2})$ as the family grows? Neither the paper's constructions nor its upper bounds answer this, and it is directly testable with the same Cayley-set method.
The Hamilton-path upper bound rests on the vertex-transitive-graph lemma, so the same counting argument should apply to any vertex-transitive graph whose vertices are structured objects and whose edges are 'creates a forbidden subgraph' relations; applying it to other forbidden subgraphs is a natural next step.

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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

Colleague,

I read the Byrne–Tait paper with the referee's concerns in hand, and I think the reject verdict is too hasty. The reader is right that the statement of Theorem 3 as written — M_k(S_n) = (n!)^{1/k} + O(1/log n) — is not what the proof establishes, and on a literal reading it is false. But the proof's final display concludes (n!)^{1/k+O(1/log n)}, and in that exponent form the argument goes through. The point the reader missed is that the factor n!/n^n from Egorychev–Falikman is not dropped; it is absorbed into the exponent error. For a = c n^{1/k}, per(M) ≥ a^n n!/n^n = c^n n^{n/k} e^{-n+O(log n)}, and writing this as (n!)^{1/k - D/log n} with D = 1 - 1/k - log c > 0 is legitimate because (n!)^{-D/log n} = e^{-D n}. So the central claim, reasonably interpreted, is not unsupported. The theorem statement needs to be corrected, and the line 'per(M) ≥ a^n n!/n^n ≥ a^{n-O(n/log n)}' invites exactly the misreading the referee made; it is only true with a sufficiently large implicit constant. That is a genuine presentation flaw, not a fatal one.

What is actually new: the permanent embedding from an S_k-set in a group of order n into S_n is a fresh idea, and the conjugacy construction of S_2-sets in S_n × S_n (Theorem 4, Proposition 2) is clean and exact. The digraph section is the strongest part. Theorems 7 and 8 give order-of-magnitude determinations of the minimum-semidegree Turán functions for F_k and C_{ℓ,ℓ}, which are new and use the Cayley graph connection to S_k-sets sensibly. The directed Erdős–Simonovits counterexample is a nice application of the comparison between those two theorems. Corollary 1's improvement on even-cycle-creating Hamilton paths is also solid. I did not find circularity or citation problems; the Odlyzko–Smith and Egorychev–Falikman uses are appropriate.

Soft spots beyond the Theorem 3 statement: the lower bound only gives M_k ≥ (n!)^{1/k-o(1)}, so calling it 'tightness of the trivial bound' oversells it; the gap is e^{Θ(n)}. Also, the proof of Theorem 3 is compressed in a way that would trip up any careful reader, and the paper would benefit from an explicit display of the Stirling computation.

Verdict: worth sending to referees after the authors fix the statement and the permanent estimate. The digraph results alone justify referee time.

Referee Report

3 major / 3 minor

Summary. The paper studies nonabelian Sidon sets S_k and S'_k in groups, with focus on the symmetric group S_n and related groups. It claims explicit constructions of large S_k-sets in S_n, S_2-sets in S_n x S_n and A_n x A_n, probabilistic constructions in 'nice' groups, and improved upper bounds on M_k(Gamma) for groups with abelian subgroups. It also connects these sets to directed extremal graph theory: it determines up to a constant factor the minimum semidegree that forces certain even cycles, improves an upper bound on Hamilton paths creating two-part cycles, and claims that a directed version of the Erdős--Simonovits compactness conjecture is false. The main theorem (Theorem 3) asserts M_k(S_n) = (n!)^{1/k+O(1/log n)} via a permanent-based construction.

Significance. If the central constructions were valid, the paper would give a substantial improvement over the previously known lower bounds for Sidon-type sets in symmetric groups, and the directed graph applications in Theorems 7 and 8 are interesting and appear to be independent contributions. The explicit nature of the constructions and the connections drawn between additive combinatorics and extremal graph theory are valuable. However, the proof of the headline lower bound for S_k-sets in S_n contains a fundamental gap: the set A' constructed by the permanent method is not shown to be an S_k-set. This undermines the paper's main advertised claim.

major comments (3)

[Section 3.2, proof of Theorem 3] The step 'By the definition of A', there exist a_1,...,a_k,b_1,...,b_k in A such that alpha_k(x)=x a_k, beta_k(x)=x b_k, etc.' is not legitimate. For pi in A', the element a_x = x^{-1} pi(x) is allowed to depend on x. In the composition alpha_1 ... alpha_k(x), the multiplier used by alpha_i is evaluated at the current point alpha_{i+1} ... alpha_k(x), which varies with x; it cannot be represented by a single fixed element a_i. Consequently the equality x a_k ... a_1 = x b_k ... b_1 does not follow, and the argument that A' is an S_k-set collapses. The lower bound M_k(S_n) >= (n!)^{1/k+O(1/log n)} is therefore not proved. This is a load-bearing error in the paper's central construction.
[Section 6, Lemma 1] Lemma 1 is false as stated for arbitrary real vectors. For K = Z_2, the vector x = (-1/2, -1/2) satisfies sum_k x_k x_{k^{-1}g} = 1/2 for every g, but x_1 = -1/2, not 1/2. The proof uses 'Thus A_1^2 = 1 so A_1 = 1' without justification; A_1 could be -1. The lemma becomes true if the hypothesis x in [0,infty)^K is added, which is the case in the application to Theorem 5, but the statement as written needs correction.
[Section 7, final paragraph] The claimed disproof of the directed Erdős--Simonovits compactness conjecture is not a consequence of Theorems 7 and 8 as stated. The notation C_{k,k} is used both for a single digraph and for the family {C_{2,2},...,C_{k,k}}. The theorems give m_0(n,F_k) = Theta(n^{1/k}) and m_0(n,C_{ell,ell}) = Theta(n^{1/2}), but the needed inequality m_0(n,C_{k,k}) = o(n^{1/2}) for the family is not proved; the available upper bound is only O(n^{1/2}). If the intended family is F_k instead, the argument must address every member of F_k, which is not done and is not immediate because F_k contains digraphs with no C_{ell,ell} subgraph.

minor comments (3)

[Throughout] The symbol C_{k,k} is used for both a single digraph and the family {C_{2,2},...,C_{k,k}}; this ambiguity is particularly confusing in the final compactness paragraph and should be resolved with separate notation.
[Section 3.2] The inequality chain per(M) >= a^n n!/n^n >= a^{n-O(n/log n)} should state explicitly that the O(n/log n) term is negative and of size Theta(n/log n); as written it invites the misinterpretation that the lower bound is n^{n/k+o(n)}, which would contradict the trivial upper bound.
[Theorem 3 statement] The statement should clarify that the O(1/log n) is in the exponent, and note that the proof gives only the lower bound n!^{1/k-O(1/log n)}, with the upper bound being the trivial n!^{1/k}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central constructions and bounds are derived from external theorems and independent arguments.

full rationale

The paper's main claims are derived from external benchmarks rather than from their own conclusions. Theorem 3 uses the Odlyzko-Smith construction of S_k-sets and the Egorychev-Falikman permanent lower bound to build S_k-sets in S_n; the verification that the lifted set A' is an S_k-set is a direct proof from the defining equations, not an assumption of the conclusion. The constructions in Theorem 4 are explicit and verified through conjugacy-counting via Proposition 2, with no fitted parameters. The probabilistic bounds in Proposition 1 use the Babai-Sos non-uniform Turan theorem and count forbidden equations directly. The upper bounds in Theorem 5 and Propositions 5-8 are independent stability arguments based on Dimovski's earlier work, not on the present paper's results. The directed graph bounds in Theorems 7 and 8 use the Odlyzko-Smith S_k-sets and an explicit graph construction G_{ell,m}, with counting via the BEST theorem in Corollary 1. The only self-citation is reference [9], which is cited as background for previously known bounds and is not load-bearing for the new results. No step reduces, by construction or by self-citation, to its own input; the paper is self-contained against external theorems and benchmarks.

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

The paper relies on standard external theorems; no ad hoc axioms or invented entities are introduced. The permanent-based proof of Theorem 3 is where the fatal error occurs, not in the axiomatic assumptions.

assumptions (5)

standard math Egorychev-Falikman theorem: permanent of an n by n doubly stochastic matrix is at least n!/n^n.
Used in Theorem 3 proof to lower-bound per(M/a).
standard math Baker-Harman-Pintz prime distribution theorem, Theorem 10.
Used in Claim 1 to find primes in short intervals for arbitrary n.
domain assumption Odlyzko-Smith construction of S_k-sets in groups of order k(p^k-1) with size (p-1)/k, Theorem 2.
Provides the base S_k-set A embedded into S_n in Theorem 3.
standard math BEST theorem for counting Eulerian circuits, Theorem 11.
Used in Corollary 1 to count Hamilton cycles in G_{r,m}.
standard math Lemma 5: alpha(G) omega(G) <= |V(G)| for vertex-transitive graphs.
Used in Corollary 1 to convert an independent set of Hamilton paths into an upper bound.

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Cite this review

Pith. "Pith review of New constructions and bounds for nonabelian Sidon sets with applications to Tur\'an-type problems." pith.science (2026). https://pith.science/paper/AHKETZXT

@misc{pith2026250907750,
  author       = {Pith},
  title        = {Pith review of: New constructions and bounds for nonabelian Sidon sets with applications to Tur\'an-type problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHKETZXT}},
  note         = {Machine review of arXiv:2509.07750}
}

abstract

An $S_k$-set in a group $\Gamma$ is a set $A\subseteq\Gamma$ such that $\alpha_1\cdots\alpha_k=\beta_1\cdots\beta_k$ with $\alpha_i,\beta_i\in A$ implies $(\alpha_1,\ldots,\alpha_k)=(\beta_1,\ldots,\beta_k)$. An $S_k'$-set is a set such that $\alpha_1\beta_1^{-1}\cdots\alpha_k\beta_k^{-1}=1$ implies that there exists $i$ such that $\alpha_i=\beta_i\text{ or }\beta_i=\alpha_{i+1}$. We give explicit constructions of large $S_k$-sets in the group $S_n$ and $S_2$-sets in $S_n\times S_n$ and $A_n\times A_n$. We give probabilistic constructions for `nice' groups which obtain large $S_2$-sets in $A_n$ and $S_2'$-sets in $S_n$. We also give upper bounds on the size of $S_k$-sets in certain groups, improving the trivial bound by a constant multiplicative factor. We describe some connections between $S_k$-sets and extremal graph theory. In particular, we determine up to a constant factor the minimum outdegree of a digraph which guarantees even cycles with certain orientations. As applications, we improve the upper bound on Hamilton paths which pairwise create a two-part cycle of given length, and we show that a directed version of the Erd\H{o}s-Simonovits compactness conjecture is false.

Figures

Figures reproduced from arXiv: 2509.07750 by the authors.

**Figure 1.** The graph Gℓ,m when ℓ “ 4 and m “ 2. Assume for a contradiction that G contains a Cℓ,ℓ composed of the two directed paths x0, . . . , xℓ and y0, . . . , yℓ where x0 “ y0 and xℓ “ yℓ . By the symmetry of the j- and k-coordinates, we may assume that x0 P Vi for some 0 ď i ď ℓ ´ 2. Then V pCℓ,ℓq X W0 “ txℓ´1´i , yℓ´1´iu. Since x0 “ y0 and xℓ´1´i ‰ yℓ´1´i , the structure of GrV0 Y ¨ ¨ ¨ Vℓ´2 Y W0s guarantees that xℓ´1´i… view at source ↗

**Figure 2.** The solid lines describe a Hamilton cycle in [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

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