Extremal problems for pairs of triangles
Abstract
A convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. There are eight pairwise nonisomorphic cgh's consisting of two disjoint triples. These were studied at length by Bra{\ss} (2004) and by Aronov, Dujmovi\'c, Morin, Ooms, and da Silveira (2019). We determine the extremal functions exactly for seven of the eight configurations. The above results are about cyclically ordered hypergraphs. We extend some of them for triangle systems with vertices from a non-convex set. We also solve problems posed by P. Frankl, Holmsen and Kupavskii (2020), in particular, we determine the exact maximum size of an intersecting family of triangles whose vertices come from a set of points in the plane.
Keywords
Cite
@article{arxiv.2010.11100,
title = {Extremal problems for pairs of triangles},
author = {Zoltán Füredi and Dhruv Mubayi and Jason O'Neill and Jacques Verstraëte},
journal= {arXiv preprint arXiv:2010.11100},
year = {2020}
}
Comments
Supersedes version 1 significantly. Only a new reference were given to version 2 (and the linear term in Thm 6 was also calculated). 25 pages, 16 figures