Author:
Keszegh Balázs,Pálvölgyi Dömötör
Abstract
AbstractWe prove that the number of tangencies between the members of two families, each of which consists of n pairwise disjoint curves, can be as large as $$\Omega (n^{4/3})$$
Ω
(
n
4
/
3
)
. We show that from a conjecture about forbidden 0–1 matrices it would follow that this bound is sharp for so-called doubly-grounded families. We also show that if the curves are required to be x-monotone, then the maximum number of tangencies is $$\Theta (n\log n)$$
Θ
(
n
log
n
)
, which improves a result by Pach, Suk, and Treml. Finally, we also improve the best known bound on the number of tangencies between the members of a family of at most t-intersecting curves.
Funder
ELKH Alfréd Rényi Institute of Mathematics
Publisher
Springer Science and Business Media LLC
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics
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