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PreprintCC BY 4.0arXiv

A somewhat sure note on an un-Schur problem

  • Swaroop Hegde
  • Hitesh Kumar
  • Pratibha
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Abstract

Parczyk and Spiegel initiated the study of an anti-Ramsey multiplicity variant of Schur's theorem and proved that the maximum fraction of Schur triples that can be rainbow in a $3$-coloring of { 1, … ,n } is bounded asymptotically between $0.4$ and $0.66364$. Furthermore, they conjectured that their lower bound is optimal. We disprove this conjecture and prove new bounds. In particular, we show that the maximum fraction of rainbow Schur triples that can be rainbow in a $3$-coloring of { 1, … ,n } lies between 9/22 and 8/15 asymptotically. Moreover, we study the problem in the general k-color setting and establish new non-trivial bounds.

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Cite

BibTeXRIS
BibTeX
@misc{hegde2026somewhat,
  title = {{A somewhat sure note on an un-Schur problem}},
  author = {Swaroop Hegde and Hitesh Kumar and Pratibha},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18474},
  eprint = {2609.18474},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18474},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A somewhat sure note on an un-Schur problem
AU  - Swaroop Hegde
AU  - Hitesh Kumar
AU  - Pratibha
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18474
UR  - https://arxiv.org/abs/2609.18474
AB  - Parczyk and Spiegel initiated the study of an anti-Ramsey multiplicity variant of Schur's theorem and proved that the maximum fraction of Schur triples that can be rainbow in a $3$-coloring of { 1, … ,n } is bounded asymptotically between $0.4$ and $0.66364$. Furthermore, they conjectured that their lower bound is optimal. We disprove this conjecture and prove new bounds. In particular, we show that the maximum fraction of rainbow Schur triples that can be rainbow in a $3$-coloring of { 1, … ,n } lies between 9/22 and 8/15 asymptotically. Moreover, we study the problem in the general k-color setting and establish new non-trivial bounds.
ER  -