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PreprintarXiv licencearXiv

Almost Empty Monochromatic Triangles With Many Colors

  • Bhaswar B. Bhattacharya
  • Sandip Das
  • Sk Samim Islam
  • Aashirwad Mohapatra
  • Saumya Sen
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Abstract

Given integers c≥ 2 and s≥ 0, let M3(c,s) denote the least integer such that every set of at least M3(c,s) points in the plane, no three on a line, colored with c colors, contains a monochromatic triangle with at most s interior points. Further, let λ3(c) be the least integer such that M3(c,λ3(c))<∞. \citet{colorempty} proved that, for every c≥ 2, ⌊(c−1)/2⌋ ≤ λ3(c)≤ c−2. Later, \citet{cravioto2019almost} improved the upper bound to c−3, for c≥ 4. In this paper, we refine their argument to obtain the following asymptotic improvement: λ3(c) ≤ c−√(clog c)+o (√(clog c) ), for all sufficiently large c. We also show that every c-coloring of a sufficiently large Horton set contains a monochromatic triangle with at most ⌊ (c−1)/2 ⌋ interior points. This shows that the aforementioned lower bound on λ3(c) is sharp within the class of Horton sets. We conclude with a conjecture on the large-color asymptotics of λ3(c).

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Cite

BibTeXRIS
BibTeX
@misc{bhattacharya2026almost,
  title = {{Almost Empty Monochromatic Triangles With Many Colors}},
  author = {Bhaswar B. Bhattacharya and Sandip Das and Sk Samim Islam and Aashirwad Mohapatra and Saumya Sen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12325},
  eprint = {2609.12325},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12325},
}
RIS
TY  - UNPB
TI  - Almost Empty Monochromatic Triangles With Many Colors
AU  - Bhaswar B. Bhattacharya
AU  - Sandip Das
AU  - Sk Samim Islam
AU  - Aashirwad Mohapatra
AU  - Saumya Sen
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.12325
UR  - https://arxiv.org/abs/2609.12325
AB  - Given integers c≥ 2 and s≥ 0, let M_3(c,s) denote the least integer such that every set of at least M_3(c,s) points in the plane, no three on a line, colored with c colors, contains a monochromatic triangle with at most s interior points. Further, let λ_3(c) be the least integer such that M_3(c,λ_3(c))<∞. \citet{colorempty} proved that, for every c≥ 2, ⌊(c−1)/2⌋ ≤ λ_3(c)≤ c−2. Later, \citet{cravioto2019almost} improved the upper bound to c−3, for c≥ 4. In this paper, we refine their argument to obtain the following asymptotic improvement: λ_3(c) ≤ c−√(clog c)+o (√(clog c) ), for all sufficiently large c. We also show that every c-coloring of a sufficiently large Horton set contains a monochromatic triangle with at most ⌊ (c−1)/2 ⌋ interior points. This shows that the aforementioned lower bound on λ_3(c) is sharp within the class of Horton sets. We conclude with a conjecture on the large-color asymptotics of λ_3(c).
ER  -