New results on the odd- and unique-Ramsey numbers

@misc{das2026results,
  title = {{New results on the odd- and unique-Ramsey numbers}},
  author = {Shagnik Das and Ying-Sian Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.07322},
  eprint = {2605.07322},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.07322},
  note = {CC BY-NC-ND 4.0},
}

TY  - UNPB
TI  - New results on the odd- and unique-Ramsey numbers
AU  - Shagnik Das
AU  - Ying-Sian Wu
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2605.07322
UR  - https://arxiv.org/abs/2605.07322
AB  - The odd-Ramsey number r_odd(n,H) of a graph H is the minimum number of colors needed to edge-color K_n so that in every copy of H some color occurs an odd number of times, and the unique-Ramsey number r_u(n,H) is the corresponding notion in which some color is required to occur not only an odd number of times but exactly once. In this paper, we address three questions from previous papers. We show r_odd(n,K_s,t)> n^1/(s/2+1/(2⌊ t/8 ⌋)) when s≤ t and s is odd and t is even, which is log-asymptotically tight when s is fixed and t→∞. Next, we consider the odd-Ramsey number when the host graph to be edge-colored is a super-Dirac graph, and show that the smallest minimum degree that forces a non-trivial odd-Ramsey number of Hamilton cycles is n/2+1. Finally, we show that r_u(n,C_n)> n/4, which leads to a polynomial gap between r_odd(n,C_n) and r_u(n,C_n).
ER  -