TY - GEN
T1 - Zarankiewicz’s Problem via ϵ-t-Nets
AU - Keller, Chaya
AU - Smorodinsky, Shakhar
N1 - Publisher Copyright: © Chaya Keller and Shakhar Smorodinsky.
PY - 2024/6/1
Y1 - 2024/6/1
N2 - The classical Zarankiewicz’s problem asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph Kt,t. Kővári, Sós and Turán proved an upper bound of O(n2− 1t ). Fox et al. obtained an improved bound of O(n2− d1 ) for graphs of VC-dimension d (where d < t). Basit, Chernikov, Starchenko, Tao and Tran improved the bound for the case of semilinear graphs. Chan and Har-Peled further improved Basit et al.’s bounds and presented (quasi-)linear upper bounds for several classes of geometrically-defined incidence graphs, including a bound of O(n log log n) for the incidence graph of points and pseudo-discs in the plane. In this paper we present a new approach to Zarankiewicz’s problem, via ϵ-t-nets – a recently introduced generalization of the classical notion of ϵ-nets. Using the new approach, we obtain a sharp bound of O(n) for the intersection graph of two families of pseudo-discs, thus both improving and generalizing the result of Chan and Har-Peled from incidence graphs to intersection graphs. We also obtain a short proof of the O(n2− d1 ) bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp O(n logloglognn ) bound for the intersection graph of two families of axis-parallel rectangles.
AB - The classical Zarankiewicz’s problem asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph Kt,t. Kővári, Sós and Turán proved an upper bound of O(n2− 1t ). Fox et al. obtained an improved bound of O(n2− d1 ) for graphs of VC-dimension d (where d < t). Basit, Chernikov, Starchenko, Tao and Tran improved the bound for the case of semilinear graphs. Chan and Har-Peled further improved Basit et al.’s bounds and presented (quasi-)linear upper bounds for several classes of geometrically-defined incidence graphs, including a bound of O(n log log n) for the incidence graph of points and pseudo-discs in the plane. In this paper we present a new approach to Zarankiewicz’s problem, via ϵ-t-nets – a recently introduced generalization of the classical notion of ϵ-nets. Using the new approach, we obtain a sharp bound of O(n) for the intersection graph of two families of pseudo-discs, thus both improving and generalizing the result of Chan and Har-Peled from incidence graphs to intersection graphs. We also obtain a short proof of the O(n2− d1 ) bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp O(n logloglognn ) bound for the intersection graph of two families of axis-parallel rectangles.
KW - VC-dimension
KW - Zarankiewicz’s Problem
KW - pseudo-discs
KW - ϵ-t-nets
UR - http://www.scopus.com/inward/record.url?scp=85195515708&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.SoCG.2024.66
DO - https://doi.org/10.4230/LIPIcs.SoCG.2024.66
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 40th International Symposium on Computational Geometry, SoCG 2024
A2 - Mulzer, Wolfgang
A2 - Phillips, Jeff M.
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 40th International Symposium on Computational Geometry, SoCG 2024
Y2 - 11 June 2024 through 14 June 2024
ER -