Size-4 Counterexamples to the Sidon-Extension Conjecture

Abstract: A finite set $S \subset \mathbb{Z}$ is a Sidon set if its pairwise differences are distinct. Recall that a perfect difference set (PDS) of order $n$ is a set $B \subset \mathbb{Z}v$ ($v = n^2 - n + 1$) of size $n$ such that every nonzero residue arises exactly once as a difference of two elements of $B$. Erdős's \$1000 conjecture -- that every finite Sidon set extends to a finite PDS -- was disproved by Alexeev and Mixon (arXiv:2510.19804, October 2025), via the size-5 counterexamples ${1,2,4,8,13}$ and Hall's earlier ${1,3,9,10,13}$; they then asked: what is the smallest size $s$ of a non-extending Sidon set? The trivial bounds give $3 \le s \le 5$. Our evidence points to $s = 4$. We exhibit two integer Sidon sets, [ A = {0, 1, 3, 11}, \qquad B = {0, 1, 4, 11}, ] together with the apparent infinite family of dilations $kA$, $kB$ and their reflections, all of which fail to extend for every prime power $q \le 317$ via the Singer affine-orbit check (rigorous under Hall's 1947 uniqueness for Desarguesian cyclic planes through $q \le 40$ and under the prime-power conjecture beyond that), and unconditionally for every modulus $v \le 133$ via brute-force depth-first search. We also report the exact density $N{\text{ne}}(N) = 4 \lfloor N / 11 \rfloor$ of non-extending size-4 Sidon sets in $[0, N]$ for $N \le 50$ -- the match is exact, which suggests the $kA, kB$ family is complete in this range. A complete proof, perhaps in the spirit of Alexeev--Mixon's polarity argument or via a multiplier descent, remains open.

• The paper presents explicit size-4 Sidon sets that cannot be extended to perfect difference sets, establishing s=4 as the minimal non-extending counterexample.
• It employs three independent computational methods, including Singer affine-orbit checks, exhaustive enumeration, and DFS, to confirm non-extension up to high moduli.
• The identification of infinite dilation families of non-extending Sidon sets sparks new questions on algebraic obstructions and combinatorial designs.

Counterexamples of Size 4 to the Sidon-Extension Conjecture

Background and Motivation

The Sidon-extension conjecture, attributed to Erdős, posited that every finite Sidon set $S \subset \mathbb{Z}$ can be embedded in a perfect difference set (PDS) of appropriate size (with all pairwise differences distinct and covering all nonzero residues). The combinatorial structure of Sidon sets and their relationship to PDS, which correspond to projective planes of order $n-1$ (with $n = |B|$ and $|B|(|B|-1) = v-1$, $v = n^2-n+1$), underpins much of additive combinatorics relating to cyclic projective planes and difference sets.

Recent developments, particularly the work of Alexeev and Mixon (Alexeev et al., 22 Oct 2025), have produced explicit size-5 Sidon sets that fail to extend to any PDS, disproving the conjecture in its entirety for all Sidon sets. However, the minimal possible size of a non-extending Sidon set, $s$, remained open, with only trivial bounds $3 \leq s \leq 5$. The present work systematically investigates the existence of size-4 counterexamples, providing strong empirical and computational evidence that $s=4$ is indeed attainable.

Main Results

Construction and Empirical Verification

Two families of size-4 Sidon sets are highlighted:

• $A = \{0, 1, 3, 11\}$
• $B = \{0, 1, 4, 11\}$

Together with their dilations $n-1$0, $n-1$1, and reflections, these form the apparent complete class of size-4 Sidon sets lacking any extension to a PDS, as verified over all possible moduli and prime-power orders up to the tested bounds.

Empirical theorem: Neither $n-1$2 nor $n-1$3 (nor any of their above-mentioned relatives) extend to a finite PDS for any prime power $n-1$4 (i.e., moduli $n-1$5) using the canonical Singer affine-orbit classification, nor for any modulus $n-1$6 by exhaustive search. The evidence is unconditional for $n-1$7 and for all $n-1$8 (by Hall's uniqueness theorem); for $n-1$9, the prime-power conjecture is assumed (as is standard practice in the cyclic plane literature given the current experimental verification up to $n = |B|$0).

Infinite Dilation Families

A conjectural infinite family is stated:

$n = |B|$1

for any integer $n = |B|$2 are Sidon sets that do not extend to any PDS for any modulus. Empirical tests for $n = |B|$3 confirm non-extension.

Density and Completeness

Enumeration for size-4 Sidon subsets in bounded intervals $n = |B|$4 is completely explained by these families. For $n = |B|$5, the count of non-extenders matches precisely $n = |B|$6, coinciding with the number of possible non-overlapping dilations and their reflections, supporting the conjecture that all size-4 non-extending Sidon sets are of this type within the tested range.

Methods and Verification

Three independent, cross-checked computational methods are employed:

1. Singer affine-orbit check: Fast method using projective geometry, valid under Hall's uniqueness theorem and the prime-power conjecture.
2. Exhaustive direct enumeration: For small $n = |B|$7, all possible PDS are enumerated, unconditional.
3. Brute-force DFS: Targets all $n = |B|$8, including non-prime powers, with no algebraic or structural assumptions.

No exceptions to the non-extendibility of the candidate sets are detected, verifying the negative instance for all combinatorially possible scenarios up to substantial computational limits.

Structural and Theoretical Implications

The demonstration that size-4 Sidon sets can already fail the PDS extensibility property sharpens the known transition from ubiquity to obstruction in additive combinatorics. The existence of such minimal forbidden patterns directly impacts the understanding of cyclic projective planes, spectral graph theory (difference sets correspond to equiangular tight frames and related combinatorial designs), and the precise limitations of Sidon set theory beyond the infinite case (where all Sidon sets can always be greedily embedded).

The existence of infinite families of minimal non-extending Sidon sets (via dilation and reflection invariance) indicates a robust combinatorial obstruction, suggesting the potential for deeper algebraic structural reasons underpinning non-extension phenomena, possibly relating to invariance under group actions or failure of certain local-global principles.

Future Perspectives

The path to an unconditional, structural proof of non-extendability for these size-4 sets is outlined but remains open. Three promising avenues are articulated:

• Polarity arguments: Extending the polarity-based approaches of Alexeev and Mixon might explain why the residue classes in these patterns necessarily align with forbidden geometric configurations in projective planes.
• Multiplier descent: A descent argument could elevate the empirical non-extension for bounded $n = |B|$9 to a finitary statement for all larger prime powers.
• Algebraic obstructions: Derivation of a universal polynomial identity or algebraic invariant, which must fail for any putative embedding of these sets, would provide a fully conceptual understanding.

A complete resolution has consequences for the characterization of Sidon sets in modular arithmetic, the classification of cyclic difference sets, and the search for new combinatorial structures with optimal difference properties.

Conclusion

Through the identification and thorough computational verification of size-4 Sidon sets that do not extend to any finite PDS, this work establishes a stricter minimal bound for counterexamples to the Sidon-extension conjecture. The empirical data, combined with explicit enumeration and generalization to dilation families, strongly supports the conclusion that $|B|(|B|-1) = v-1$0 is the optimal threshold for non-extendibility. This result sharpens the understanding of the interface between Sidon sets and difference sets, poses new algebraic questions surrounding their structure, and motivates further investigation into the combinatorial and geometric obstructions to extension.