A Structure Theorem for Pseudo-Segments and Its Applications
We prove a far-reaching strengthening of Szemerédi’s regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such graphs can be partitioned into a bounded number of parts of roughly the same size such that almost all of the bipartite graphs between pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every n-vertex simple topological graph with no k pairwise disjoint edges has at most n(log n)^O(log k) edges.
Regularity lemma
pseudo-segments
intersection graphs
Mathematics of computing~Combinatorics
59:1-59:14
Regular Paper
https://arxiv.org/pdf/2312.01028
Jacob
Fox
Jacob Fox
Department of Mathematics, Stanford University, CA, USA
Supported by NSF Award DMS-2154129.
János
Pach
János Pach
HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Rényi Institute, Budapest. Supported by NKFIH grant K-176529, Austrian Science Fund Z 342-N31 and ERC Advanced Grant "GeoScape."
Andrew
Suk
Andrew Suk
Department of Mathematics, University of California San Diego, La Jolla, CA, USA
Supported by NSF CAREER award DMS-1800746, and NSF awards DMS-1952786 and DMS-2246847.
10.4230/LIPIcs.SoCG.2024.59
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https://arxiv.org/abs/2312.01028
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Jacob Fox, János Pach, and Andrew Suk
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