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.