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Graham conjecture on small sets in abelian groups

This paper employs a recursive approach to prove that subsets of size up to 20 (or 22 for zero-sum subsets) in generic abelian groups are sequenceable, significantly improving upon the previous bound of 9 and advancing the resolution of Graham's conjecture and the related CMPP conjecture.

Original authors: Simone Costa, Stefano Della Fiore, Mattia Fontana, Lluís Vena

Published 2026-03-24

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CC BY 4.0

Original authors: Simone Costa, Stefano Della Fiore, Mattia Fontana, Lluís Vena

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a bag of unique, magical stones. Each stone has a number written on it, and these numbers belong to a special world called an "Abelian Group" (think of it as a playground where you can add numbers together, but the order you add them doesn't change the result).

The big question mathematicians have been asking for decades is: Can you line up these stones in a specific order so that as you count them one by one, the running total never repeats a number, and never hits zero (unless it's the very end)?

If you can do this, the set of stones is called "sequenceable."

This paper is like a team of detectives (Costa, Della Fiore, Fontana, and Vena) who have cracked a major case regarding how many stones you can have in your bag before this becomes impossible.

The Big Mystery: Graham's Conjecture

Back in the day, a mathematician named Graham guessed that no matter how many stones you have (as long as they aren't zero), you can always find a magic order to line them up.

The Problem: For a long time, we could only prove this was true for very small bags of stones (up to 9 stones).
The Recent Breakthrough: Recently, other mathematicians proved it works for huge bags of stones, but only if the playground is very specific (like a prime number universe).
The Gap: What about medium-sized bags (say, 10 to 20 stones) in any kind of playground? That was the missing piece.

The Team's Solution: The "Merge" Trick

The authors didn't try to solve the whole problem at once. Instead, they used a clever recursive strategy, which we can call the "Merge and Shrink" technique.

Imagine you are trying to arrange a messy line of people.

The Rule: You need to find two people in the line, let's call them Alice and Bob.
The Merge: You ask them to hold hands and become a single new person, "Alice-Bob," whose value is the sum of their two numbers.
The Check: If "Alice-Bob" is a unique person (not already in the line) and isn't zero, you have successfully shrunk your problem! You now have one fewer person to arrange.
The Recursion: If you can prove that every time you shrink the group, you can eventually get down to a tiny group that we already know how to arrange, then the original big group must also be arrangeable.

The authors proved mathematically that for bags of up to 20 stones, you can always find two stones to merge without breaking the rules.

The Results: Pushing the Boundaries

Using a mix of elegant math proofs and a super-powered computer search (their "digital detective"), they pushed the known limits way further:

The General Case: They proved that for any bag of up to 20 non-zero stones, you can always find a valid order. (Before this, we only knew this for 9).
The "Zero-Sum" Case: If the stones in your bag add up to zero (like a balanced scale), the limit goes up to 22.
The "No Opposites" Case: If your bag has no pairs of stones that cancel each other out (like +5 and -5), the limit goes up to 23.

How They Did It: The Computer Tree

To prove this, they built a massive "search tree" in their computer code.

The Tree: Imagine a tree where every branch represents a different way to order the stones.
The Dead Ends: As the computer explores these branches, it looks for "forbidden collisions" (where the running total repeats).
The Certificate: If the computer finds that every possible path leads to a contradiction (like proving two different stones must be the same person), it stops and says, "Aha! This bag cannot be a counter-example."
The Power: They used a supercomputer with 128 processors to check these paths. They found that for bags up to size 20, the "dead ends" always appeared, proving that a valid order must exist.

Why This Matters

Think of this like building a bridge. For a long time, we could only build bridges for small rivers (small sets) or for massive oceans (very large sets with specific rules). This paper builds the missing bridge for the "medium rivers."

They showed that the universe of these number games is much more orderly than we thought. Even with complex rules and different types of number systems, as long as you don't have too many stones (under 20), there is always a way to line them up perfectly.

In short: They took a 50-year-old math puzzle, used a "merge" trick to shrink the problem, and used a supercomputer to prove that for small-to-medium groups, the answer is always YES, you can line them up.

1. Problem Statement

The paper addresses the Graham Conjecture (also known as the Graham/Alspach Conjecture) regarding the sequenceability of subsets in abelian groups.

Definition of Sequenceability: Let $G$ be an abelian group and $A \subseteq G \setminus \{0\}$ be a finite subset. An ordering $a_1, \dots, a_{|A|}$ of $A$ is valid if all its partial sums $p_i = \sum_{j=1}^i a_j$ are distinct. The ordering is sequencing if it is valid and $p_i \neq 0$ for all $1 \le i \le |A|-1$ . A set $A$ is sequenceable if it admits a sequencing ordering.
The Conjecture: Conjecture 1.1 states that every subset $A \subseteq G \setminus \{0\}$ of an abelian group $G$ is sequenceable.
Related Conjecture (CMPP): Conjecture 1.2 (Costa, Morini, Pasotti, Pellegrini) posits that if $A \subseteq G \setminus \{0\}$ sums to zero ( $\sum_{x \in A} x = 0$ ) and contains no inverse pairs (i.e., $\{x, -x\} \not\subseteq A$ ), then $A$ is sequenceable.
Context: While recent breakthroughs (Pham & Sauermann, 2024) proved the conjecture for sufficiently large primes $p$ using discrete Fourier analysis, the problem remains open for general abelian groups and small sets. Prior to this work, the conjecture was verified only for sets of size $|A| \le 9$ in general abelian groups.

2. Methodology

The authors employ a hybrid approach combining algebraic characterization and computational search via a recursive reduction strategy.

A. The Recursive Lemma (Theoretical Core)

The central theoretical tool is a Recursive Lemma (Corollary 2.5). It asserts that for any set $A \subseteq G \setminus \{0\}$ , either:

$A$ is already sequenceable, OR
There exist distinct elements $x_1, x_2 \in A$ such that their sum $s = x_1 + x_2$ is non-zero and $s \notin A \setminus \{x_1, x_2\}$ .

If condition (2) holds, one can construct a smaller set $A' = (A \setminus \{x_1, x_2\}) \cup \{x_1 + x_2\}$ of size $|A|-1$ . If $A'$ is sequenceable, this sequencing can be "lifted" to a sequencing of $A$ .

The authors provide two proofs for the existence of such $x_1, x_2$ :

Polynomial Method (for $\mathbb{Z}_p$ ): Using Alon's Combinatorial Nullstellensatz, they construct a homogeneous polynomial $F_k$ of degree $\frac{(k+2)(k-1)}{2}$ . They show that for $10 \le k \le 23$ , there exists a monomial with a non-zero coefficient where the degree of each variable is less than $|A|$ . This guarantees the existence of the required pair to merge.
Direct Algebraic Proof (General Groups): They characterize subsets where every pair sum lies in $A \cup \{0\}$ . Using ideas related to Kneser's Theorem, they prove (Theorem 2.4) that if every pair sum is in $A \cup \{0\}$ , then $A \cup \{0\}$ must be a subgroup. Since the conjecture is known to hold for subgroups (minus the identity), the recursive step is always possible unless the set is trivial.

B. Computational Search (Breadth-First Search)

To verify the conjecture for specific sizes $k$ , the authors developed a proof-by-contradiction algorithm:

Assumption: Assume a counterexample $A$ of size $k$ exists (i.e., no valid ordering exists).
Search Tree: The algorithm explores the space of orderings (permutations) of $A$ .
Constraint Matrix ( $C$ ): As the search proceeds, it records "forbidden" intervals. If a consecutive block of elements in an ordering sums to zero (a collision), it generates a linear relation $\sum v_i a_i = 0$ , encoded as a binary incidence vector $v$ .
Terminal Certificates: The search stops a branch if the row span of the constraint matrix $C$ $C$ over $\mathbb{Q}$ $Q$ implies a contradiction, such as:
- Zero-element: $a_i = 0$ (contradicts $A \subseteq G \setminus \{0\}$ ).
- Equality: $a_i = a_j$ for $i \neq j$ (contradicts distinctness).
- Compression: The relations imply a structure that contradicts the known sequenceability of the reduced set $A'$ (via the Recursive Lemma).
Implementation: The algorithm uses Breadth-First Search (BFS) and linear algebra (rank tests via numpy) to detect these certificates. It was run on a high-performance server (128 cores, 2 TB RAM).

3. Key Contributions and Results

The paper significantly extends the known bounds for sequenceability in general abelian groups.

Main Theorems

Theorem 1.4 (General Case): Every subset $A \subseteq G \setminus \{0\}$ $A \subseteq G ∖ {0}$ is sequenceable if $|A| \le 20$ .
- Improvement: Previous best bound was $|A| \le 9$ .
Theorem 1.5 (Zero-Sum Case): If $\sum_{x \in A} x = 0$ , then $A$ is sequenceable if $|A| \le 22$ .
Theorem 1.6 (CMPP Conjecture Case): If $\sum_{x \in A} x = 0$ and $A$ contains no inverse pairs ( $\{x, -x\} \not\subseteq A$ ), then $A$ is sequenceable if $|A| \le 23$ .

Technical Details

The authors verified that for $k \le 23$ , the recursive reduction always succeeds or a contradiction is found computationally.
They explicitly calculated coefficients for the Combinatorial Nullstellensatz polynomial for $k \in [10, 23]$ (Table 1), confirming the polynomial method works for $\mathbb{Z}_p$ in this range.
The computational search confirmed that no counterexamples exist for these sizes in generic abelian groups.

4. Significance

Resolution of Small Cases: This work closes the gap for small sets in generic abelian groups, a domain where previous results were limited to size 9.
Methodological Innovation: The combination of the Recursive Lemma (reducing the problem size) with a constraint-based BFS search (verifying the base cases) provides a powerful framework. The authors note that without the Recursive Lemma, the pure tree-algorithm would fail at size 10, consistent with prior literature.
Computational Feasibility: The paper demonstrates that sequenceability problems, often considered intractable for large $N$ , can be solved for specific structural constraints (small $k$ ) using modern computational algebra and parallel processing.
Foundation for Future Work: By establishing the sequenceability of sets up to size 23, the paper provides a robust base for further inductive proofs or for tackling the general conjecture for larger sets in specific group structures.

In summary, the paper proves that the Graham Conjecture holds for all subsets of size up to 20 in any abelian group, and up to 23 under specific structural constraints (zero-sum, no inverse pairs), utilizing a novel recursive reduction strategy backed by rigorous computational verification.