Some results on intersecting families of triangulations

Abstract

Kalai conjectured that the largest intersecting family of triangulations of an n-gon is the family consisting of all triangulations that contain a fixed ear (a chord of length $2$). What happens if we strengthen the intersecting condition to the condition that any two of the triangulations must intersect on a chord from a specified family of chords -- for example, what if any two triangulations intersect on a chord from some fixed point of the n-gon, or intersect on an ear? In this paper we verify Kalai's conjecture for families of triangulations in which any two intersect on a chord from some fixed point. We also show that any ear-intersecting family of triangulations has size at most T_n−1 +T_n−3 = T_n−1(17/16 + o(1)), where T_n−1 is the upper bound in Kalai's conjecture, namely the number of triangulations of an (n−1)-gon. This is based on an exact upper bound of T_n−1 for the size of a family of triangulations in which any two intersect on some ear from the set of all ears with one ear excluded.

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BibTeX RIS

BibTeX

@misc{leader2026some,
  title = {{Some results on intersecting families of triangulations}},
  author = {Imre Leader and Ta Sheng Tan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39116},
  eprint = {2610.39116},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39116},
}

RIS

TY  - UNPB
TI  - Some results on intersecting families of triangulations
AU  - Imre Leader
AU  - Ta Sheng Tan
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39116
UR  - https://arxiv.org/abs/2610.39116
AB  - Kalai conjectured that the largest intersecting family of triangulations of an n-gon is the family consisting of all triangulations that contain a fixed ear (a chord of length $2$). What happens if we strengthen the intersecting condition to the condition that any two of the triangulations must intersect on a chord from a specified family of chords -- for example, what if any two triangulations intersect on a chord from some fixed point of the n-gon, or intersect on an ear? In this paper we verify Kalai's conjecture for families of triangulations in which any two intersect on a chord from some fixed point. We also show that any ear-intersecting family of triangulations has size at most T_n−1 +T_n−3 = T_n−1(17/16 + o(1)), where T_n−1 is the upper bound in Kalai's conjecture, namely the number of triangulations of an (n−1)-gon. This is based on an exact upper bound of T_n−1 for the size of a family of triangulations in which any two intersect on some ear from the set of all ears with one ear excluded.
ER  -

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