Un-Schur problem and Parczyk–Spiegel conjecture

For $n\ge 1$ , let $T_n=\{(x,y,z)\in\{1,\ldots,n\}^3:x+y=z\}$ be the set of Schur triples, and for a coloring $c:\{1,\ldots,n\}\to\{1,2,3\}$ let $R_n(c)$ be the number of triples $(x,y,z)\in T_n$ whose three entries receive pairwise distinct colors. Determine the asymptotic extremal fraction $\alpha_3=\limsup_{n\to\infty}\max_c\frac{R_n(c)}{|T_n|}$ . The currently reported bounds are $\frac{9}{22}\le\alpha_3\le\frac{8}{15}$ ; the exact value is unknown.

References

Primary source

arXiv

Additional references

A somewhat sure note on an un-Schur problem — arXiv — Swaroop Hegde, Hitesh Kumar, Pratibha

extremal-combinatorics ramsey-theory

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

A new note disproves the earlier conjectured value and narrows the possible asymptotic fraction, but the exact answer remains unknown.

The problem asks for the largest asymptotic fraction of Schur triples that can be rainbow in a coloring, in the anti-Ramsey setting associated with the Un-Schur problem and Parczyk–Spiegel conjecture. The exact extremal fraction is not known.

Known results

An earlier study established asymptotic bounds $0.4$ and $0.66364$ for the $3$ -color case and conjectured that $0.4$ was optimal.
The same work developed extensions to rainbow $k$ -term arithmetic progressions.

September 16, 2026 bounds

Hegde, Kumar, and Pratibha report that the earlier conjectured extremum is false: the asymptotic fraction is bounded between $9/22$ and $8/15$ , with an analysis extending to general $k$ -colorings. This is a claimed advance from a preprint, not an independently verified resolution.

Current status (as of September 2026): The exact asymptotic optimum remains open; a September 2026 preprint claims new bounds $9/22$ and $8/15$ and refutes the earlier conjectured lower-bound optimum.

Sources

No solutions have been posted yet.