The conjecture that the Schur-degree threshold equals the Schur number

About 6 years old · traced to 2020

For $n\ge 2$ , let $S(n)$ be the $n$ th Schur number. Let $L(n)$ be the least length such that every sequence $A$ in $\mathbb N_+$ with $|A|=L(n)$ and average $\mu(A)\le n$ satisfies $\operatorname{sdeg}(\hat A)\ge n$ . The Schur-degree threshold conjecture.

L(n)=S(n-1)+1.

Equivalently, every sequence $A$ in $\mathbb N_+$ of length $|A|=S(n-1)+1$ and average $\mu(A)\le n$ satisfies $\operatorname{sdeg}(\hat A)\ge n$ . The conjecture is known for $n=2$ and $3$ , while the case $n=4$ is explicitly left open in the source.

References

Primary source

Shalom Eliahou and Pastora Revuelta, “The Schur degree of additive sets”, arXiv:2006.01502 (2020).

additive-combinatorics schur-numbers

Posted 1 month ago · edited 2 days ago

Progress summary

Refreshed 2 days ago

Claimed progress

The conjecture remains open: an unverified submission claims it fails in dimensions four and five.

The conjecture predicts that the threshold $L(n)$ equals $S(n-1)+1$ . The cases $n=2$ and $n=3$ are known, while $n=4$ was previously left open.

Known results

The conjecture is established for $n=2$ and $n=3$ ; no resolution of $n=4$ was recorded in the cited source.

Community submission (unverified), October 4, 2026

A submitted argument claims $L(4)=16$ , using a length- $15$ sequence intended to disprove the conjectured value $L(4)=14$ . It also argues that $49\leq L(5)\leq61$ , based on a length- $48$ construction. These claims have not been independently verified.

Current status (as of October 2026): The cases $n=2$ and $n=3$ are settled; the claimed counterexamples for $n=4$ and $n=5$ are unverified, and the general conjecture remains open.

Sources

CounterexampleThe conjecture predicts L(n) = S(n-1) + 1. Since S(3)=13 and S(4)=44 this predicts L(4)=14 and L(5)=45. We prove that L(4)=16 and L(5)>48 refuting the conjecture.See full solutionHide full solution

Theorem 1.2 (The degree-four threshold)

We have $L(4)=16$ . More precisely, the sequence

A_4=(3,1,3,4,3,1,3,4,3,1,3,4,3,1,3)

has length $15$ , total $40$ , and every one of its prefixes has average at most $4$ and blocksum Schur degree at most $3$ . This refutes the $n=4$ case of the Conjecture and the conjectural value $L(4)=14$ .

Theorem 1.3 (A degree-five interval)

We have

49\leq L(5)\leq61.

Let

W=(3,3,3,3,3,3,1,3,3,3,3,3,3,4)

and let $A_5$ be the concatenation of three copies of $W$ followed by six further entries equal to $3$ . Then $A_5$ has length $48$ , total $141$ , and every prefix has average at most $4$ and blocksum Schur degree at most $4$ . Hence $L(5)>48$ , refuting the $n=5$ case of the Conjecture.

The colourings and details are written down in the attached paper.

The examples were found with the help of KI.

Schur_L4_L5_2026-09-20.pdf82,654 bytesOpen